Through having no luck, perchance (3)
I believe the answer is:
per
'through' is the definition.
(I've seen this before)
'no luck perchance' is the wordplay.
'no' suggests deleting specific letters.
'luck' becomes 'chance' ('chance' can be a synonym of 'luck').
'perchance' with 'chance' taken out is 'PER'.
'having' acts as a link.
(Other definitions for per that I've seen before include "According to" , "For each or every" , "Preposition indicating by means of" , "Preposition meaning for each" .)