It really does depend on your rent.
For tax credits, presuming your DS is over 1 year old and your DP works full-time, you will get around this much:
Working tax credits:
Basic element: £1,920
Couple and lone parent element £1,970
30 hour element £790
Total = £4680
Child Tax Credits:
Family element £545
Child element £2,720
Total = £3265
Grand Total = £7945
However, your income reduces that total. The first £6420 is ignored. That means that your DH's £13000 will be treated as £6580.
For every £1 over £6420, 41 pence is deducted from your Tax Credits. So your total deduction would be £6580 x 0.41 = £2697.80
This would bring your total tax credit award for the whole year (remember though that you are already in June) to roughly £5247.20, which would be £1982.20 Working Tax Credits (£38.12 per week) payable to your DP and £3265 Child Tax Credits (£62.79 per week) payable to you. So £100 per week in total.
Housing benefit
You get allowances which are added together and deducted from your income. All calculations are weekly. The allowances for your situation (presuming you are both over 18) are:
Couple rate £112.55
Child £65.62
Family £17.40
Earnings disregard £10
Additional Earnings Disregard: £17.10
Total allowances: £222.67
Your income is totaled:
Tax Credits: £100
Wages: 224.19 (£13000 gross changed to net and calculated weekly).
Total income: £324.19
The allowances are taken away from the income and the total is deemed your 'excess':
£324.19 - £222.67 = £101.52
You are expected to contribute 65% of that towards your rent:
£101.52 x 0.65 = £65.99
So £65.99 is what you'd be expected to contribute towards your rent (roughly) each week.
Council used to use the same system, but deducted 20% as your contribution, so you would be expected to pay
£101.52 x 0.20 = £20.30 towards your Council Tax.
However, the changes in legislation have made Council Tax Relief a local benefit now, so some LAs calculate differently and you'd have to check.
So, you are likely to have:
£324.19 from CTC, WTC and wages.
£20.30 from Child benefit
Total: £344.49
You are likely to have to pay (at least) £65.99 + £20.30 towards rent and Council Tax, so a total of £86.29.
That leaves you £258.20 for utilities, food, and living costs.
Those figures are all rough, btw - don't rely on them exactly.