Indian perhaps dumping a Welsh girl? (4)
I believe the answer is:
sian
'welsh girl?' is the definition.
(common Welsh name)
'indian perhaps dumping a' is the wordplay.
'indian perhaps' becomes 'Asian' (indian is a kind of Asian).
'dumping' is a deletion indicator.
'asian' with 'a' removed is 'SIAN'.
(Other definitions for sian that I've seen before include "woman in Cardiff" , "Welsh name" .)