If your partner has been let go from his job and is available for work, then he can sign on and claim Jobseeker's Allowance or Benefit at the local Social Welfare Office. If he's unfit for work (depending on how bad his back is), he might qualify for Illness Benefit (by sending medical certs into Social Welfare), or if you're available for work, you could sign on.
If there's a delay in getting a social welfare payment, or if you need help with rent, contact the local Community Welfare Officer at the nearest health centre. S/he may also consider assistance towards a deposit for you at a new address.
When you're with the Community Welfare Officer, ask about a medical card and the back to school clothing scheme.