Yes, there is a Qualified Child Allowance which is 29.80 per week. If you and your partner are claiming Social Welfare, the allowance may be split between you. Contact your local office where you sign on and they'll tell you what they need to sort your claims.
Your mortgage supplement would not change as your income will remain the same compared to the SWA rate for your family size.
If you haven't already applied for a medical card, you should do so at your local HSE office.
You can apply for an Exceptional Needs Payment if you are having difficulty getting larger items for the baby (e.g. cot or buggy). It's not an automatic entitlement, each case will be considered individually. Contact your local CWO for details.
And congrats on the new baby!