It's a tricky situation, particularly if you are a lone parent (I'm not sure if this is the case but your OP would suggest it might be). A safety net of savings is particularly difficult to relinquish when it's your only back up if things go awry, so I understand why you want to hold on to it whilst you get yourself back on your feet.
If you are not working, you can still pay into a pension up to £2880 each tax year, which will be topped up to £3600 by the government in tax relief. This is not counted as deprivation of assets by DWP.
You can also pay off any outstanding debts.
If you have worked in the last 2-3 years have you checked if you might be eligible for the new style job seekers allowance? It is not dependent on savings.
https://www.citizensadvice.org.uk/benefits/help-if-on-a-low-income/jobseekers-allowance-jsa/before-you-claim-jsa/check-if-you-can-get-jsa/
If you find yourself unable to claim I would focus on the following:
Ensure your savings are receiving the best possible rates of return - move your money regularly to keep up with the best rates available.
Cut down all your short term outgoings and costs as much as possible - look at bank statements to see what can be trimmed.
Look to see if you might be eligible for other benefits that are not affected by savings to see if you might be eligible - such as new style ESA, PIP.
Try to bring in some additional income whilst at home - selling unwanted stuff on ebay; set up an extra dummy bank account so you can pursue bank switching bonuses; matched betting (not gambling) are the most lucrative and most flexible options, particularly as you have some savings to use for this - it will give a regular return greater than interest on your savings.
Depending on the age of your child - if they have a particular interest or skill - look to see what grants are available to support this. They are sometimes available for things such as music, sport, IT but it takes a bit of tracking down.
If you are well enough and looking to apply for work in the future, and your child is of school age or you have childcare available think about the following:-
Try to gain some voluntary work even if only a couple of hours a week. Often if you have been away from the job market for a while it's difficult to obtain references and a voluntary role is an ideal way of solving this as well as building confidence in your own abilities.The see if there are any training opportunities that might be available to you.
Try to gain some additional qualifications online that might stand you in good stead for job hunting. There are many free courses online.
If you have no support network around you, make this a priority to focus on. For instance, if you are a person of faith in some way, connect to a welcoming church (or other place of worship). It can provide a valuable supportive community. If that's not right for you, then look to other neighbourhood groups that might provide similar.
It sounds as if you have had a challenging time, so I hope things improve for you.