Are search warrants limited to evidence for a specific crime?
YES Different Acts have slightly different wording but they all require the police to suspect an offence, for example s.8(1) of the Police and Criminal Evidence Act 1984:
(1) If on an application made by a constable a justice of the peace is satisfied that there are reasonable grounds for believing —
(a) that an indictable offence has been committed; and
(b) that there is material on [the] premises ... which is likely to be of substantial value (whether by itself or together with other material) to the investigation of the offence;
(e) he may issue a warrant authorising a constable to enter and search the premises...
s.15(6)(b) requires all search warrants to state what is being searched for:
(6) A warrant—
(b) shall identify, so far as is practicable, the articles or persons to be sought.
Searches are also limited to the extent required for the purpose for which the warrant was issued: s.16(8) - meaning a search warrant for a stolen car does not authorise looking in drawers, cupboards etc unlike one issued for, say, a stolen mobile phone. Once the items have been found (or not as the case may be) officers must vacate the premises unless there is another lawful reason to remain there.
Would this marijuana be admissible in court for a charge of possessing that marijuana?
If an officer finds evidence of an unrelated offence - either during a search under a warrant or is lawfully on the premises for any other reason - then s.19 is available to seize the items which may open a new investigation:
(1) The powers conferred by subsection ... (3) ... below are exercisable by a constable who is lawfully on any premises.
(3) The constable may seize anything which is on the premises if he has reasonable grounds for believing—
(a) that it is evidence in relation to an offence which he is investigating or any other offence; and
(b) that it is necessary to seize it in order to prevent the evidence being concealed, lost, altered or destroyed.
If anyone was arrested for an indictable offence in connection with that other evidence (such as possession of a controlled drug) then this would trigger the non-warrant seach of premises power at s.32 to look for related evidence:
(2) ... a constable shall also have [the] power in any such case —
(b) if the offence for which he has been arrested is an indictable offence, to enter and search any premises in which he was when arrested or immediately before he was arrested for evidence relating to the offence.
Meaning although the original warrant is no longer extant, a new search can begin for evidence of the newly discovered drugs offence.