The question isn’t why it’s constructed like that; I’ve just sort of absorbed the fact that it is.
The question: is it then correct to construct a phrase like “Dw i’n gwybod y liciwn i” or “Dw i’n {what’s the verb for feel?} y liciwn i” or the like?
Come to think of it, I might as well ask: why is it like that? 
Yes, it is correct 
The answer to the question why is, as always with languages: just because.
‘Dw i’n teimlo y liciwn i esbonio rhywbeth’
If you join two sentences, and the second sentence does not have a present version of bod (ydw, wyt, mae, etc.), you can use ‘y’ to link them - if the second sentence is a negating sentence, you use ‘na’
A-ha! Thanks for explaining it, and also for giving the negative. 
Having just run into the “na” (as in ‘na ddylwn i ddim’), I really must thank you for mentioning it here, or I’d be completely at sea.
Hi @louis, really helpful tip, thank you! Does it still work in the sentence ‘she told me that I have time’ (L1 Challenge 23) which still seems to use ‘bod’ rather than ‘y’ even though the following clause doesn’t use a present tense of bod: ddudodd hi wrthaf fi bod gen i amser i wneud o.
Hi @sara-28,
In your example the English doesn’t have a present tense form of ‘to be’ but the Welsh equivalent does, which is why you need ‘bod’ and not ‘y’.
The English ‘I have’ is rendered in Welsh as ‘there is with me’. If you split the example you have given taking away the ‘ddudodd hi wrthaf fi’ you are left with, as a standalone phrase, ‘mae gen i amser i wneud o.’ That ‘mae’ becomes ‘bod’ when it becomes a subordinate clause.
Unit 11 of Gareth King’s Intermediate Welsh explains this with clarity and some exercises to practice it.
Thank you! That makes perfect sense now - I hadn’t realised the second does indeed use a present form of bod.