In pursuit of time, becoming sick (7)
I believe the answer is:
tailing
'in pursuit' is the definition.
'tailing' can be an answer for 'pursuit' (tailing is a kind of pursuit). I'm not sure about the 'in' bit.
'time becoming sick' is the wordplay.
I cannot quite understand how this works, but
'time' could be 't' (abbreviation) and 't' is found in the answer.
'sick' could be 'ailing' (similar in meaning) and 'ailing' is found within the answer.
This accounts for all the letters.
This explanation may well be incorrect...
'of' acts as a link.
Can you help me to learn more?
(Another definition for tailing that I've seen is " Following".)