Write off American, increasingly injecting this out of hospital (8)
I believe the answer is:
amortise
'write off' is the definition.
(I've seen this before)
'american increasingly injecting this out of hospital' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'increasingly' becomes 'more'.
'injecting' indicates putting letters inside.
'out of' is a deletion indicator.
'hospital' becomes 'H' (abbreviation for hospital on maps).
'more' enclosing 'this' is 'morthise'.
'morthise' with 'h' taken away is 'mortise'.
'a'+'mortise'='AMORTISE'
(Other definitions for amortise that I've seen before include "Liquidate gradually" , "Gradually write off initial costs" .)