Stay with president who's not all there (4)
I believe the answer is:
bide
'stay with' is the definition.
'president who's not all there' is the wordplay.
'president' becomes 'biden' (Joe Biden).
'who's not all there' means to remove the last letter.
'biden' with its last letter taken off is 'BIDE'.
(Other definitions for bide that I've seen before include "Remain (archaic)" , "Remain or stay (poetically)" , "Remain; await (time)" , "Stay in a certain place" , "Dwell, of old" .)