FWTW, according to CNN:
Republican Rep. Patrick McHenry of North Carolina will now temporarily lead the House of Representatives after the speaker's position was vacated on Tuesday.
McHenry, who is a top ally of Kevin McCarthy, was appointed speaker pro tempore.
His name was on a list McCarthy was required to give to the clerk in case of a vacancy.
The speaker pro tempore, which is the official title, can only recess the House, adjourn the chamber and recognize speaker nominations.
So, if that is right, it looks like the powers of the pro-tempore speaker [who was essentially pre-selected by the outgoing one] are fairly limited (for now). Possibly if the situation persists as such, the House might feel a need to increase those powers at his disposal.
Also FWTW, Deschler's precedents note this distinction between an elected and a designated pro-tempore Speaker:
An elected Speaker pro temporeis more than a ‘‘stand-in’’ Speaker.Indicative of this is the requirement that he swear a new oathupon his entering the office ofSpeaker pro tempore.Moreover, an elected Speakerpro tempore assumes a muchgreater scope of authority fromThe Speaker than a designatedSpeaker pro tempore. [...]
He must,however, be authorized by theHouse to perform certain dutieseven though he has been electedby the House, and not simply designated by The Speaker.Examples of the kinds of duties,powers, and functions assumed byan elected Speaker pro temporefrom The Speaker include: administering the oath of office to newMembers; (1) appointing conferees; (2) appointing committees towait on the President and to inform him that the session’s workis completed; (3) or that a quorumof both Houses is ready to receivehis state of the Union message; (4)signing enrolled bills and jointresolutions during the adjournment of the House; (5) declaring recesses during a session; (6) andpresiding at a joint session of theCongress.(7)
However, McHenry has been designated rather than elected, so his powers appear indeed to be more reduced because of that.
There's another House Manual that notes that
When the Office of Speaker is vacant, the Member actingas Speaker pro tempore under rule I section 8(b) may exercise such authorities of the Office as may be necessary and appropriate pending the election of a Speaker or Speaker pro tempore.
However, it doesn't elaborate with any examples of what might be "necessary and appropriate" in such circumstances. Consequently Reuters concludes that
The acting speaker's duties are vague, according to a guide to the chamber's rules and procedures.
Anyhow, for practical purpose in this instance, they quote:
Republican Representative Kelly Armstrong told reporters that McHenry's main task will be to "get us a new speaker." Anything further, he said, would spark a move to oust McHenry.
Although I'm not exactly sure what makes Armstrong's view determinative in this case.
Also FWTW, there's the diametrically opposed opinion of some commentators that the speaker pro-tempore has full powers subject to the "support of the conference" (which is not too clear what it means--perhaps not getting vacated themselves).
Josh Huder, a senior fellow at the Government Affairs Institute at Georgetown University, wrote on X, formerly known as Twitter. "There's not [a] forcing mechanism for a new election, nor are there any overt restrictions on the power the pro tem would wield. The support of the conference would dictate the durability of this."
As that opinion doesn't reference any rules or laws, I'm guessing it comes down to a different interpretation of "necessary and appropriate" in the preceding discussion. (In doing so, it does seem to cast aside most if not all precedents from Deschler's though.)