Tory’s opening dispute involves hospital project (5)
I believe the answer is:
throw
'project' is the definition.
(I know that project can be written as throw)
'tory's opening dispute involves hospital' is the wordplay.
'opening' says to take the initial letters.
'dispute' becomes 'row' (row is a kind of dispute).
'involves' means one lot of letters goes inside another.
'hospital' becomes 'H' (abbreviation for hospital).
The initial letter of 'torys' is 't'.
't'+'row'='trow'
'trow' going around 'h' is 'THROW'.
(Other definitions for throw that I've seen before include "puzzle" , "shot-put event?" , "Hurl, fling" , "lose on purpose" , "Form (pot) on a wheel" .)