I hope I’ve found the right part of the forum to ask this:
Why is it: Wela i chdi benwythnos nesa and Dw i ddim yn siwr, os wela i chdi penwythnos nesa?
I’m hoping to use my Welsh for the first time this St David’s day - I don’t get much opportunity in Auckland (NZ).
Welcome aboard, and it’s a good place and things to ask.
You have an answer on its way 
Nobody knows why 
But Gareth King in his ‘Modern Welsh’ p.246 para 403 writes: "time adverbs - primarily those which indicate ‘when’, ‘how often’, or ‘for how long’ - undergo soft mutation,.so ‘wela I chdi benwythnos nesaf’ is ‘correct’, but that does nor mean that if you say ‘penwythnos nesaf’ anyone will notice…
I’ve also been having this question, as I can’t seem to figure out any reasonable explanation for why it gets said with the soft mutation sometimes and without it other times. I’ve been noticing the same thing with “I enjoyed” or “did you enjoy” - sometimes it seems to be “wnest ti fwynhau” or “wnes i fwynhau” but sometimes it’s “wnest ti mwynhau” and I can’t figure out which is correct.
Why is it said both ways? Because people forget! I know about the adverbial time thingy, but can I remember in the heat of the moment? Of course not. The only person who cares is my Welsh tutor, when I’m writing it down, and even then it’s only because I’m paying him to care. No need for a Welsh tutor by the way, I’m just a masochist and want to improve my writing.