The Prime Minister, and other government ministers, naturally have the time they can dedicate to constituency matters limited by the duties of their roles in government. However, as an important part of an MP's duty is to represent their constituents, this is never completely ignored.
Anecdotally, the website of the current Prime Minister, Boris Johnson, allows members of his Uxbridge constituency to book a ten-minute appointment, and states that he holds advice surgeries across the area. The 'News' page of his website also seems to document most of his constituency appearances.
In general, tasks such as responding to correspondence, following up and providing briefs on local issues, local media management, and organisation of constituency events is handled by a constituency office, and a large proportion of correspondence from constituents is not be seen personally by the Prime Minister.
In her article for the Institute for Government, Becoming Prime Minister, Dr Catherine Haddon looks at the approach of other recent Prime Ministers, finding a mixture of approaches:
Being Prime Minister also has an impact on constituency work. A new
prime minister can choose how to undertake the role, but it must be
managed against all their other commitments. Prime ministers in recent
history have rarely spent more than a day every other week in the
constituency, depending on foreign travel and other commitments. For
many it was a once-a-month activity. Thatcher had a monthly surgery in
Finchley. For David Cameron it was one Friday a month, plus quite a
lot at weekends. For Theresa May constituency time was restricted to
Saturdays – between twice a month and more frequent. It also partly
depends on how far away they were. Blair travelled to Sedgefield far
less frequently: it was several weeks before he first visited as Prime
Minister. One of the difficulties for prime ministers is keeping
focused while being dragged from issue to issue.
Unfortunately, as constituency interactions are held in the Prime Minister's capacity as a member of parliament, freedom of information requests to the Prime Minister's Office on this subject have failed, suggesting that this sort of anecdotal data could be all that is available.
With regard to the balance between the Prime Minister's role as a constituency MP, and their role in government, we can look to the Ministerial Code.
Ministers in the House of Commons must keep separate their roles as
Minister and constituency Member
[...]
Ministers are provided with facilities at Government expense to enable
them to carry out their official duties. These facilities should not
generally be used for Party or constituency activities.
This separation of a Prime Minister's responsibility to their constituency, as well as their responsibility to enact manifesto commitments can often lead to apparent contradictions - for example, much was made of David Cameron's correspondence with Oxfordshire county council protesting against cuts to public services, which, the council leader pointed out, came as a result of the Conservative Party's manifesto commitment to eliminating the deficit.
The trade-off for constituents, however, is of course representation at the highest level of government; instead of one's MP writing to a Minister, or asking a ministerial question in the Chamber, the Prime Minister can bring this up directly in Cabinet, and a letter from the Prime Minister's Office is likely to hold more weight than any other MP.
