Please persevere with tax credits. You should definitely be entitled to some, I can't understand why you think you're not?
They don't take any savings into account, or maintenance (other benefits do, and universal credit will when it comes in), so its purely based on actual income you earn.
I think the limit for a single parent is around £19k for one child, and around £26k for 2, so would be more for 3. Certainly I'm a single parent with 2 kids earning just under £24k, and I get a fair amount each month (would be around £200 I think, but I get more as one of my DC gets PIP, so you get an enhanced payment). Anyway, even a couple of hundred pounds should make a difference, so definitely keep trying with them. The calculator on "entitled to" is generally pretty good.
Make sure you're getting your single persons council tax discount if you're not already. Thats worth 25%. Some of the water companies do a discount I believe if you have 3 kids or more, or are on certain benefits (in fact thats just reminded me to phone, as my colleague gets a discount for getting child tax credits, and she earns more than me, so need to check that out!).
I know its come about through awful circumstances, but being mortgage free is definitely a good position to be in, as it means you don't need so much coming in each month to be able to survive, and there is no risk of losing your house. I'm trying to get my mortgage paid of ASAP, I live a fairly frugal lifestyle to do this. I've got some issues with illness that may or may not be fixed after a big op in the summer, so at least if I can get my house paid off then its one thing less to worry about.
Being a single parent is hard, I've done it for 14 years. When you have sole responsibility for your kids, its far more difficult to build up a career and progress through it, unlike the NRP who doesn't have any of those childcare worries. I'm lucky to be in a secure job, but I have put any hopes of promotion on hold, and have taken positions because they have fitted in with the kids rather than because I was necessarily interested in them. And if it wasn't for tax credits paying my childcare when I started my job, I would never have been able to afford to work, as £800 a month in nursery fees just wouldn't have been possible.
It does get easier, and I wish you all the luck in the world.