The intended reference is surely to Section 1078 of the National Defense Authorization Act for Fiscal Year 2013, which is about "Voice of America" and other media services produced by the Department of State for non-U.S. audiences.
Under the United States Information and Educational Exchange Act of 1948, also known as the Smith-Mundt Act, the Secretary of State was authorized
to provide for the preparation, and dissemination abroad, of information about the United States, its people, and its policies, through press, publications, radio, motion pictures, and other information media, and through information centers and instructors abroad.
This law did not explicitly authorize distribution of that material domestically, other than saying it would be available on request to members of Congress and in certain other circumstances. The policy of not making it generally available in the U.S. was hardened into law in the Foreign Relations Authorization Act of 1972, saying
Any such information (other than "Problems of Communism" which may continue to be sold by the Government Printing Office) shall not be disseminated within the United States, its territories, or possessions, but, on request, shall be available in the English language at the Department of State, at all reasonable times following its release as information abroad, for examination only by representatives of United States press associations, newspapers, magazines, radio systems, and stations, and by research students and scholars, and, on request, shall be made available for examination only to Members of Congress.
Subsequent tweaks carved out many exceptions for other specific content, and arranged for deposition of material in the National Archives. In 1985, a new provision was inserted to say that apart from certain enumerated exceptions,
no funds authorized to be appropriated to the United States
Information Agency shall be used to influence public opinion in the
United States, and no program material prepared by the United
States Information Agency shall be distributed within the United
The developing statute law reflects a balance of concerns; it was felt that Americans should know what their government is saying to foreign audiences, but should not be the target of that official messaging. These additional provisions are not a departure from the position of 1948, so much as they are trying to set out the boundaries more clearly in the light of experience.
The 2013 law, codified within 22 U.S.C. §1461-1464b, made some changes to the domestic regime. It authorizes post-2013 material to be made generally available to the U.S. public on request, potentially subject to a fee: there is no longer a requirement that the requestor has to be a scholar/journalist/etc. It also amends the "no domestic distribution" section to account for that change. The bar on influencing domestic political opinion remains, as can be seen from the current codification.
The present situation is that much post-2013 material is available online (which is "on request", in that you have to go to the appropriate website). There is a system for requesting permission to reuse this material, including within the U.S.; for example, you might do this if you were making a documentary and wanted to show some VoA footage. Per policy, there is a possibility for providing
program materials, prepared for dissemination abroad, to domestic broadcasters aimed at foreign diaspora communities as part of the Agency's foreign policy mission.
Subject to various criteria, this allows broadcasting material within the U.S., not by the federal government itself but by some commercial organization - if that is still compliant with the prohibition on influencing domestic political opinion, based on the way the material is used. The broadcaster is also obliged to credit the producing agency, so it should be reasonably clear when a video segment (for example) has this origin.
In answer to the question "Does H.R. 4310 allow the US government to use propaganda against citizens?", the situation is something like: it makes it possible for a third party to rebroadcast existing foreign-targeted propaganda for a domestic audience, in certain circumstances. It does not allow the government to broadcast or promote it domestically, and does not allow the government to produce it with a domestic audience in mind.
An important note is that all of this legislation only applies to the foreign-audience material produced under the auspices of the State Department (under various agency names at different times). It does not cover broadcasting or communication by other parts of the federal government.
A less important note is that the journal "Problems of Communism" mentioned in the 1972 Act is now called "Problems of Post-Communism", and published by Taylor & Francis rather than by the U.S. government.