Hi Cathal,
The short answer is no. You are cohabiting (from what I read in your post) so no. If you were living apart, you could both then claim the one parent tax credit (which I was yesterday informed by Revenue if earning 50k a year, s worth approx 50 ish euro a week). Other than that Child Benefit and the Early Childcare Supplement are paid to your partner.
If your income is low/ beneath a certain amount you can apply for family income supplement - I am not sure of the limits etc.
The system here is messed up - when not married you can't share/swap tax credits - but yet your income will be taken into consideration for social welfare etc etc