In pursuit of time, becoming sick (7)

'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 see how this works, but
'time' could be 't' (abbreviation) and 't' is present in the answer.
'sick' could be 'ailing' (similar in meaning) and 'ailing' is found within the answer.
No letters remain.
This may be the basis of the clue (or it may be nonsense).

'of' is the link.

Can you help me to learn more?