You can compete for your 2nd claim club if your 1st claim club is not involved in the competition. However, you can't compete for your 2nd claim club in an open competition - whether your 1st claim club is involved or not.
The best way to clarify is with an examples.
1: Say your your 1st claim club is a member of the Surrey X/C league and your second claim club is in the Kent X/C league. Both leagues are 'closed' so you can run for either club.
2: Both clubs are in the same X/C league - you can only compete for your 1st claim club, even if they are in different divisions.
3: An 'open' road race has a team event - you can only score for your 1st claim club, regardless of whether they have a team entered.
4: Bear in mind however, that leagues and associations don't have to allow 2nd claim members. For example, you can run in the Surrey X/C league as a 2nd claim member, but not in the Birmingham X/C league .
Hope that makes sense.