Catch, having run into touch (8)
I believe the answer is:
contract
'catch' is the definition.
(to catch an illness is to contract it)
'having run into touch' is the wordplay.
'having run' becomes 'r' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'into' is an insertion indicator.
'touch' becomes 'contact' (contact is a kind of touch).
'r' placed into 'contact' is 'CONTRACT'.
(Other definitions for contract that I've seen before include "Document setting out terms of agreement" , "type of bridge" , "Shorten" , "Undertake" , "four spades, perhaps" .)