Have you made the application for rent supplement in writing, or was it just a conversation with the CWO? If you haven't completed application forms, then you should do this first.
As part of your application, include the additional information (recent unemployment, baby due, deposit paid and contract signed prior to unemployment) pertinent to your claim.
If you are refused (you should get a written refusal), you have the right of appeal.
For information, the maximum rent limits set by Dept of Social and Family Affairs for Cork are - Couple with no children 153 per week, with 1 child 175 per week. Rent supplement is generally not payable where the rent exceeds these amounts. However exceptions may be made, so its worth your while persuing the claim and the appeal if it comes to it.
Could you sub-let a room in the house to help with the rent?
Does the landlord own other properties with cheaper rent, he/she might allow you to transfer?