I am confused by the apparent inconsistency of soft mutation with to think. My revision guide gives these sentences as correct:
Beth rwyt t’n / dych chi’n feddwl am y ffilm?
Rwyt ti’n meddwl …
Beth mae’n meddwl am y ffilm?
It appears that only ti and chi mutate the verb after beth, and there is no mutation when not a question, or is there a different reason? Nothing is explained in the book
I know that when the ‘beth’ is the object (rather than the subject) of the sentence, the following verbnoun often has a soft mutation, but I can’t explain it further so I’m going to tag @garethrking
Well now…
The first thing to say about the three examples quoted by @berwick-curtis from his revision guide:
is that the third one is simply wrong - it should be Beth mae’n feddwl am y ffilm, by the same principle that makes the first example correct.
No, neither the ti nor the chi are mutating the verb - for a start there is an intervening 'n anyway; the reason for the mutation here is that 'n feddwl is really 'n ei feddwl, with the ei ‘echoing’ the beth because it’s the object of the verb (Siaron correctly spots as usual) but appears before it rather than after. It’s sort of like saying What does he think (it) of the film?
Well there’s a reason for that. 
Many thanks for the reply. My third example had a typo - should have read " …mae e’n …and …mae Owen yn meddwl …
Would that make any difference?
This distinction between the “you” and the “he/she” version appears in the revision booklet for the Welsh GCSE exam board and is repeated in several “beth” questions in different places??
The suspense is getting to me now.
So, Beth is short for Pa beth (what thing)
I’m pretty sure mae e’n, mae’n and Mae Owen yn all amount to the same thing.
I’m still pretty sure that all Mae… sentences will be: Mae x yn meddwl, and all Beth mae questions will be: Beth mae x yn feddwl?
What did the original course book manual have for these?
If there is a ‘Beth’ at the beginning then you need the soft mutation for the reasons given by Gareth above - beth mae e’n (ei) feddwl?
If it’s just a statement about someone thinking, e.g. Mae Owen yn meddwl with no beth then there’s no soft mutation
Though in casual, everyday speech you’ll hear people ignoring the rule and saying things like Beth mae fe’n meddwl? at times 
I think the connecting “yn” is causing the soft mutation. (this is the 'n).
You are right in thinking that yn can cause soft mutation, but one exception is that verbnouns are not mutated after yn / 'n.
The explanation that Deborah gave is correct: Technically, there is an ei present which causes the mutation, and that mutation is kept, even if the word itself is dropped.
This happens in other cases, as well, for example:
Wyt ti wedi gweld (fy) nghar? - Have you seen my car? – Sometimes the possessive particle fy is dropped in speech, but the nasal mutation is kept.