If your doctor says only part-time at the moment, then don't even attempt to work longer hours.
Believe me, your health is far more important than money at the end of the day. I've been a single parent and know it's possible to live for a few years (certainly struggling!) without a job.
Things WILL get better.
I'm 58 and terminal and still 'on sick', but had semi-retired anyway --took a reduced pension--and was only doing 2 days a week.
I'm actually saving money--don't need work clothes, or petrol.
money to get to and from work!
My job is still 'open', I've been replaced temporarily until July and my doc has just given me a sick note for 6 months--if I even live that much longer!
You're younger so it's obviously harder for you, but do remember your health must come first--for your own AND your children's sakes

Brenda