I agree with you OP I think it can and as sad as you think it must be for your DC it must be equally if not more so upsetting for you and your DH as well.
I can remember what it was like not being able to say yes to very much when I was a single mum to 4DC not receiving any maintanance and the limited benefits I was entitled to being mucked about and honestly for me it was bloody soul destroying and a very depressing way to live.
Have you used the Governments online calculator to check your getting every single thing that your entitled to if you haven't give it a try you never know you may be missing out on something that your entitled to.
With days out do you have any friends or family that would be happy to do a car share or car pool with you so the cost of petrol for days out is shared?
With the football clubs your DC want to join do you have family or friends that buy them presents for birthdays/Christmas or Easter maybe, would they chip in together to pay for the clubs as a present for your DC instead of an actual present to unwrap.
Like a PP mentioned have a look online for free bikes,check out ebay,freecycle,Gumtree and Preloved(there's most probably tons more that I don't know about so have a good look around)also think about putting a call out for any free bikes people you know might be getting rid of on facebook,I know some people wouldn't want anyone knowing that they needed or wanted something for free but tons of my friends that are better off are always putting up things for free or asking for things for free and are over the moon when they can help someone else out or someone else can help them out.
These are most probably things you've already done but just incase you haven't have a look through all of your incomings and outgoings and see if there's something your missing,check out what companys your with for your energy supplies,your house phone(if you have one)mobile provider,tv packages(if you have them)internet,credit cards,insurances everything.We recently changed our gas and electric suppliers and we're going to save over £100 a year and my DH rang up Sky and said we were going to leave them so they've knocked our bill down by nearly £30 a month(saving us nearly £360 a year)and we've been given Sky Q,we have an extra box now and we get netflix so you can shave some money of your bills by shopping around.
A longer term plan I know but is there any chance of pay increase with either of your jobs?Either through extra training that you could access through your work or a change in job or taking on extra hours.
If one of you is working part time could you do some babysitting on the side for friends/family/neighbors and in return they could take your DC out(they're treat)in the school holidays or at the weekends.
I hope things start improving for you soon OP 