Don’t start run with snake (5)
I believe the answer is:
adder
'snake' is the definition.
(type of snake)
'don't start run' is the wordplay.
'don't start' means to remove the first letter.
'run' becomes 'ladder' (both can mean a rip in tights).
'ladder' with its first letter taken off is 'ADDER'.
'with' acts as a link.
(Other definitions for adder that I've seen before include "He's making an extension" , "Venomous sort" , "Reptilian" , "danger in the countryside" , "a poisonous type" .)