The Looking Glass - this is an online magazine from of Trinity College, Dublin. It is currently looking for submissions for children's or YA writing, deadline 26th September. See here and past issues. They will accept poetry, short stories, plays, novel extracts, and also illustrations and comic strips.
The Caterpillar - this is a hard copy, quarterly magazine of children's writing (7 -11 years apporoximately) from the same publishers as The Moth. They accept poetry and short story, see submission details here
The Caterpillar also have a poetry competition running currently with a deadline of 31st March 2015. There is a prize of €1,000 for the winning poem for children aged 7 - 11. And, on the topic of children's writing, here are some tips from the writer Sheena Wilkinson at the writing.ie website
Oh, and I've just noticed another competition on the website Writing.ie A.M. Heath Irish Children's Prize 2014 deadline 20th October, win €1,000 and an agent.