Births in Kilkee would be registered in Kilrush. I don’t see a birth for a Christina Madigan in 1920 in the index to Irish civil registration. Nearest is Oct – Dec 1922 Vol 4, page 210. There was also a Mary Christina Madigan Oct – Dec 1915 Vol 4, page 224.
Best matches for Mary Madigan were Jul – Sept 1920 Vol 4, page 314 & Margaret Mary Madigan Jul – Sept 1922 Vol 4, page 203.
You could order up some or all of these certificates to see if the father was Patrick. You can order a photocopy of the certificate from GRO Roscommon for €4 (euros). http://www.welfare.ie/en/Pages/Apply-for-Certificates.aspx
You have to download and print off the form. Then either post or fax it back to them. You can’t e-mail it to them. However if you want them to e-mail the cert to back to you, they will do that, so tick the relevant box.
Put the reference details on the form (anywhere). Don’t worry about leaving some boxes blank. As long as GRO have the location, name(s), year, quarter (where there is one), volume and page number they should find it.
This was the only Patrick Madigan I could see in the Kilkee area in 1911. At that time he was living in the townland of Dough:
http://www.census.nationalarchives.ie/pages/1911/Clare/Kilkee/Dough/361998/
If you can track the right birth certs for Mary & Christina, that’ll give you the parents names. That will then allow you/us to search for the marriage and from that we should get Patrick’s father’s name and occupation which will tell us whether the 1911 entry is the right family.