"The Welsh Government is in Caerdydd."
Translation:Mae Llywodraeth Cymru yng Nghaerdydd.
No. Llwywodraeth Cymru is a proper noun which translates as 'The Government of Wales' - no need for y in the Welsh.
(In fact, Llywodraeth Cymru, the administrative organisation which is 'the Welsh Government', is spread across a number of sites in Wales. Cynulliad Cenedllaethol Cymru, 'the Welsh National Assembly', is the political chamber which meets in the Senedd building in Cardiff. I think I've got that right...)