How likely are we to get a mortgage?

[Hmsxoxxx]

Hi! I'm wondering if anyone on here could give us some advice regarding getting a mortgage.
Me and my partner have decided we want to start saving for a house deposit but I quite frankly have no idea where to start.
We get about 30k a year between us but I only work part time (just had a baby) and I'm on a zero hour contract and have been there over 2 years. we will only have about 10k for a house deposit but are hoping to use the help to buy scheme. I'm just wondering if anyone has managed to get a mortgage on a zero hour contract or if I would be. Better off looking for a new job with a contract even if this will long it out as I won't have been there for long?
Thanks!