In U.S. law, there are several ways that a non-profit organization's failure to conform to its stated charitable purpose is enforced, and how non-profits, more generally, are supervised in the U.S. legal system.
The question also contains some misapprehensions about which government officials and agencies are involved in dealing with the issues it discusses (i.e. the role of the secretary of state), so I address those incorrect assumptions as well.
Also, this question was tagged "criminal law", but in fact, criminal law plays a very small part in the regulation of this aspect of conduct by non-profits.
Private State Court Litigation By An Internal Party
First, someone who is a participant in the organizational structure of the non-profit, such as a member or director of the non-profit can bring a lawsuit seeking to compel it to conform to its mission. Generally speaking, the judgment of the board of directors is given broad deference in determining what a non-profit must do to comply with its stated purpose, but this deference is not absolute. In particular, a court will often intervene if it appears that people connected to the non-profit are engaging in self-dealing transactions for their personal benefit to the detriment of this purpose. This is a violation of their legal duties to the non-profit entity.
Usually, the remedy would be (1) an injunction, ordering the non-profit to change something that it does pursuant to a court order, (2) an order for someone who has improperly received personal benefit to disgorge the benefit that they have received to a charity, or (3) an order judicially dissolving the non-profit entity because it cannot be made to serve its purpose in a way that a court can supervised in a practical manner.
Much less frequently, a non-profit that is finding it difficult to fulfill its original mission due to a change in circumstances, can apply to a court for permission to reform and amend its original mission statement and purpose in a manner that is feasible in light of the current reality, while as close to the original purpose as is feasible, under a doctrine known as the cy pres doctrine.
For example, if a charity was formed to assist people who are suffering from small pox, a disease that has now been eradicated, the charity could apply to a court to reform its purpose to allow it to assist anyone who is suffering from an infectious disease.
Most of these lawsuits, by the way, involve the "equity" element of a court's jurisdiction, rather than the portion of a court's authority known as its "legal" authority.
These lawsuits would never be criminal in nature.
Supervisory Actions By State Attorneys-General
Second, the non-taxation oriented government official who has primary supervisory authority over non-profit entities is the attorney-general of the state in which the non-profit is organized. This reflects the common law conception of attorneys-general as representatives for "the People" generally, even when particular individuals in the general public have no real particularized interest in whether a particular non-profit is conforming to its stated purpose that is different from any other member of the general public (i.e. most members of the general public don't have "legal standing" that permits them to bring suit when this is true).
This intervention would usually employ remedies similar to those of a private lawsuit by someone involved in the organization, rather than criminal penalties, with the only difference being that the attorney-general has standing to sue notwithstanding the fact that the attorney-general is not a participant in the private non-profit entity.
A state attorney-general, or a local district attorney (in states where local district attorneys are part of different government entities than the state attorney-general as they are in Ohio), could bring criminal charges in cases, for example, of embezzlement of charitable funds for personal benefit. But criminal charges would almost never be available to enforce a failure of a non-profit to carry out its mission faithfully.
The state attorney-general, rather than the U.S. attorney-general has responsibility for this because in the U.S., the organization and conduct of non-profit organizations in areas other than tax law, is a question of state law and not question allocated to the federal government (with some narrow exceptions for a handful of charities organized under federal law and, for example, for non-profits charged with managing employee fringe benefit programs which are comprehensively regulated by a federal statute known as the Employee Retirement Income Security Act a.k.a. ERISA).
The Secretary of State of a state's involvement in non-profit entity law, in contrast, while important, is strictly ministerial and bureaucratic. It simply maintains the public records showing the dates of formation and dissolution of non-profit organizations, copies of their organizational documents, and their current contact information. The Secretary of State's office has no substantive involvement in how non-profit organizations conduct themselves. If a state secretary of state received a complaint about a non-profit organizations failure to conform to its governing documents, the secretary of state would either ignore this complaint or forward it to the relevant bureau of the state attorney-general's office.
Federal Tax Law Supervision
Third, the federal Internal Revenue Service is charged with determining that non-profit organizations entitled to an exemption from federal taxes and whose donors received a charitable income tax deduction, and/or gift and estate tax exclusion, conform to the requirements of federal tax law to be eligible for this tax classification. The IRS compares the substance of non-profit organizational documents to the requirements of federal tax law, monitors and enforced the requirement that non-profit organizations file an annual tax return on IRS Form 990, and both pro-actively and in response to complaints from the general public, investigates potential violations of federal tax requirements for maintaining non-profit tax status in the day to day operations of a non-profit.
The failure of a non-profit to carry out its stated charitable purpose as set forth in its governing documents is not in and of itself a violation of federal tax laws, if the non-profit's activities are still consistent with some charitable purpose, even if it is a charitable purpose other than the one stated. But, if the non-profit engaged in non-charitable activities that benefit private individuals or entities in a manner inconsistent with its charitable purposes, this is a violation of federal tax laws.
The IRS is charged with representing the general public's interest from a tax perspective in having non-profits comply with non-profit tax laws that have the effect of reducing the amount of tax that is collected from organizational activities. But it is not charged with protecting the general public from activities that aren't consistent with a non-profit's stated purpose and may mislead the general public, but don't impact the federal tax base.
Criminal enforcement of these tax laws mostly involves false statements about particular facts on tax returns filed with the IRS for the purposes of reducing U.S. income tax collection, or for money laundering purposes.
For example, if someone tells the IRS on Form 990 that a non-profit spent 90% of the funds received as donations to make donations to a cancer hospital that doesn't exist, when that person actually took those funds and used them as a personal salary without reporting it as compensation income, the IRS would probably bring a tax fraud prosecution in that kind of case.
But, if the IRS learned that a non-profit was actually engaged mostly in lobbying, or mostly in operating a for profit business venture for the benefit of shareholders in the entity, usually, the IRS would just revoke the entity's non-profit tax status, rather than bringing a criminal lawsuit against people involve in the non-profit entity.
Fraud On Donors And In Product Endorsements
Private citizens, under both state law and federal law, local government district attorneys, state attorneys-general, and various federal government agencies such as the Fair Trade Commission, and the federal Justice Department (including local U.S. attorneys), all have overlapping authority to bring legal action related to misleading advertising.
This could take the form of fraud in connection with representations made to potential consumers of products sold or endorsed by a non-profit stating that it meets certain standards, or to donors to a non-profit acting in reliance upon the belief that the funds donated will be used to further a stated charitable mission.
Usually, these actions would be civil lawsuits, seeking money damages or injunctive relief directed at stopping misleading representations from being made by the non-profit, rather than criminal penalties.
Criminal enforcement of misrepresentations by a non-profit predominantly involve a fact pattern where someone represents that they are raising money or property for a charitable purpose, while intending from the start to use that money or property solely for their own non-charitable personal benefit.
For example, a typical case that would give rise to criminal charges would involve someone raises money for a cancer hospital that doesn't exist, has never even tried to comply with federal tax law for non-profits, and intends from the start to use the money instead to gamble and pay for strippers and food at a big party in Las Vegas, and to buy luxury cars for themselves.
But, in contrast, if an organization was actually formed under the relevant laws related to non-profits for the purpose of endorsing products that were "organic" but, in practice, cut corners by not auditing claims of people seeking their endorsement that their products were really "organic" and by endorsing products that didn't fully or substantially comply with their stated "organic" standards knowing that fact, this would usually only give rise to civil lawsuits rather than criminal charges.
In part, this reflects deferences to the private judgment of private organizations regarding their charitable purposes. Instead, our legal system largely leaves the question of whether a non-profit is faithfully carrying out its mission to the judgment of individual donors and product purchasers based upon the reputation of the organization in question and their own personal research into the reputability of that organization. Basically, our legal system assumes that judges and juries have no special expertise in making those determinations. There are, indeed, non-profit organizations that do almost nothing but help the public determine whether other non-profits are really living up to their stated purposes in a faithful and efficient manner, to address that issue.
This kind of legal action, however, has already been explored in other questions by the same questioner, so I won't develop this part of the answer more fully in this answer to avoid duplication. This question, as I read it (with an eye towards avoiding duplication), primarily concerns fidelity of a non-profit to its own internal mission, rather than the remedies related to the veracity of representations it has actively made to third-parties to induce them to take action to buy products or to make donations to it.