After pressure, priest beginning to succumb, but getting her own way (10)
I believe the answer is:
prevailing
'succumb but getting her own way' is the definition.
I can't judge whether this definition defines the answer.
'after pressure priest beginning' is the wordplay.
'after' means one lot of letters go next to another.
'pressure' becomes 'p' (scientific abbreviation).
'priest' becomes 'rev' (short for reverend).
'beginning' becomes 'ailing' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'p' put next to 'rev' is 'prev'.
'prev'+'ailing'='PREVAILING'
'to' acts as a link.
Can you help me to learn more?
(Other definitions for prevailing that I've seen before include "Generally accepted" , "ruling" , "victorious" .)