That hospital avoided getting medical gear caught in part of machine (6)
I believe the answer is:
tappet
'part of machine' is the definition.
(part of an engine)
'that hospital avoided getting medical gear' is the wordplay.
'hospital' becomes 'h' (abbreviation for hospital on maps).
'avoided' suggests deleting specific letters.
'getting' indicates putting letters inside.
'medical gear' becomes 'PPE' (personal protective equipment).
'that' with 'h' removed is 'tat'.
'tat' placed around 'ppe' is 'TAPPET'.
'caught in' is the link.
This may not be right. Some or all of it may belong to another bit of the clue.
(Other definitions for tappet that I've seen before include "Moving part in an internal combustion engine" , "Apt pet for valve operator" , "motor part" , "Lever to transmit motion from one part to another, in car engine say" , "piece of mechanism" .)