And the trigger is the word “i”, rather than the “mae gen i” construction.
For the OP and any newer learners, the shorthand reason is that “i” always causes soft mutation in the word that follows it, if that word begins with a soft-mutatable letter.
The longer reason is more interesting, and more complicated (although only slightly).
If Gareth King is reading this thread, I’ll leave it to him to explain. 
Indeed it is. The longer reason is that the i is the notional subject, and the subject in Welsh is followed by the Soft Mutation. This is why Mae gan Elwyn gi bach also has that ci mutated to gi - Elwyn is the notional subject (he’s the one possessing the little dog!).
Gan does also cause a soft mutation: “gan fod” for example.
That makes brilliant sense! He is the subject! Elwyn has a little dog! Diolch yn fawr!!!
Although actually Anthony that’s a bad example, because the fod there is not mutated by the gan, but by the fact that it’s the bod/ fod contruction meaning ‘that…’. In other words, bod can also appear after gan:
gan fod e’n mynd yfory - because he’s going tomorrow
gan bod hi’n mynd yfory - because she’s going tomorrow
gan bod nhw’n mynd yfory - because they’re going tomorrow
and indeed
gan mod i’n mynd yfory - because I’m going tomorrow
You are right, of course, about the preposition gan causing SM - this kind of example:
albwm newydd gan Gôr Rhyngwladol y Smyrffiaid - a new album by the International Choir of Smurfs
I’m a little disappointed that TGK would listen to smurf choirs. Sigh. Feet of clay and all that…
I am so sorry, but why does gan…e have fod and gan …hi or nhw have bod? My instinct would be fod for hi and bod for e or o and nhw!
I can’t give you the technical explanation that Gareth can, but it’s the same pattern as -
fy mod i
dy fod ti
ei bod e/o
ei fod hi
ein bod ni
ein bod chi
eu bod nhw
but I don’t know where those are in the course, so maybe you haven’t come across them yet.
Edit - now I think about it, in the ‘gan’ construction, the fy/dy/ei etc is ‘there’ but left out (if you see what I mean) - gan (fy) mod i - so, they would follow the same pattern, wouldn’t they! doh!
Aren’t they “blocked” by the “ei” and “eu”?
I don’t know if Gareth maybe mistyped but what you listed is what i expected with bod for e, o and nhw and for hi. Gareth had them the other way round.
oops, no, I mistyped - ei fod e/o and ei bod hi. definitely. 
still not got the hang of this multitasking thing! should engage brain fully before letting fingers loose on keyboard 
No, it’s the ei and eu that determine the mutation - then they are (often) dropped, leaving (in the case of ei masculine) the SM on its own. So gan ei fod e becomes gan fod e
Yes you can, if ti helps you to use it early on. As with any language “pair”, the “translations” are often approximate, because the two languages work differently. It’s useful to have these ideas of equivalents, so mae is the equivalent of is, but don;t hold them too literally, otherwise you’ll get confused and stressed when the English “is” suddenly becomes yw / ydy, or when mae is “are” as well.
I won’t write more, because trying to explain it is far more confusing than the reality of using it will be. If you stick with learning patterns with SSiW, and having little moments of “Oh, that’s how it works. That’s interesting” will make you a more relaxed Welsh speaker far earlier than trying to work out the internal translations of patterns.
Enjoy the stuff that is blindingly word-for-word obvious, and reallyenjoy the stuff that doesn’t follow an obvious translated pattern, because that is the stuff that will show you are mastering Welsh, not a Welsh translation of English!
Thank you for this
My original post was that a school had Mae as I have and what little knowledge I had indicated that this was wrong
This goes alongside an observation in another school where cwc was used for cook.
@aran @Iestyn @garethrking is anything to be done to correct schools teaching the next generation -er - somewhat incorrectly?
Nope!
Cwcio is a common alternative to coginio in the north, I believe.
During tea break at my Welsh class tonight we were talking about Welsh in schools. In some schools cael ’ is being taught as ‘I have’ as in ‘I have a car’! Getting away from this thought helped me understand Welsh so much. Instead of: ‘Mae gen i gar’ they are taught ‘dw i’n cael car’ and apparently they will ‘correct it later’. What is going on in these schools?
Perhaps they are reverse engineering from dwi wedi cael?
Or perhaps they’re just wrong.