Yes, for Gawne and similarly derived names but the Welsh Gwynne is derived from Gwyn
"white, fair, blessed".
eg:
Gwynne — derived from Welsh gwyn "white" or "fair hair" or "fair complexion"; it was first used as a personal name. Gwinnett, Gwyn, Gwynett, Gwynn, Gwynne, Gwyyns, Wyn, Wynn, Wynne.

GAVIN m English, Scottish, Welsh
Late medieval form of GAWAIN

GAWAIN m Welsh, Welsh Mythology, English
Possibly from the Welsh name Gwalchgwyn which meant "white hawk"...

GWYN m Welsh
Means "white, fair, blessed" in Welsh.

GWYNEDD f & m Welsh
From the name of a region in Wales, which probably derives from Welsh gwyn "white, fair, blessed".

GWYNEIRA f Welsh
Means "white snow" from the Welsh element gwyn "white, fair, blessed" combined with eira "snow".

GWYNETH f Welsh
Either a feminine variant of GWYNEDD or else possibly derived from Welsh gwynaeth "happiness".

GWYNFOR m Welsh
Derived from the Welsh element gwyn "white, fair, blessed" combined with mawr "great, large".

GWYNN m Welsh
Variant of GWYN

I am not sure about Gawain though as there may also be a mythological 'smith' link here. It may have been a pun on 'white hawk' and 'smith' as the story tellers of the middle ages were fond of hiding hidden meanings in their tales. I am not really an expert but that's the impression I get.

Jane