In the case of the title of the article you provided the link to, gClubanna is eclipsed because the na which comes before it is in the genitive plural, is that right? "Day of the Clubs"?
I was at a bit of a loss with this. I play FOR my local football team, but I play WITH a golf club (the stick, not the institution). I guess I could play WITH my local football team, it's just not normally something I would say. The distinction is useful. If I wanted to say I play WITH the club (the stick) in Irish, would that be leis an gclub too? If I wanted to say I play FOR a club (institution) is there a way to make that unambiguous?
Because it comes after the preposition le and the singular definite article. For many prepositions such as ar, ag and le, you eclipse a singular definite noun (ar an gcathaoir, ag an mbord, etc).
Unfortunately, it's not consistent - there are prepositions that don't eclipse in this way.