The sentence is: My mouth is shut tight.
My translation: My vengegie widdi Kie undin
The translation given: My vengegie widdie Guy undin.
Question: Have I missed something? I don’t remember “weddi” as being a word that normally will cause a word to mutate. But is the kie/guy change due to the vowel at the end of “weddie”?
Nope, you haven’t missed anything - that’s a good spot. But as usual, it’s nothing to worry about 
The “weddi kie” you’re hearing is wedi cau, and the “weddi guy” is likely to be wedi’i gau. There’s a grammatical-type explanation as to why the latter mutates, but unless you’re specifically interested, it’s probably best not to go too much into the details - your response is fine.
The difference in meaning between the two is pretty slight anyway - you could think of it as being like the difference between “my mouth has shut” and “my mouth is shut”. And seeing as how mutations in natural speech can be somewhat fluid anyway, I pretty much guarantee that your response would not only be understood but pass completely unnoticed
As the sentence uses 'is shut" rather than “has shut” your expliantion makes sense.
Diolch yn fower