Clare, whom I work with, is doing the London marathon this year.
The Clare (whom) I work with is doing the London marathon this year.
Object pronouns can be deleted in restrictive relative clauses:
The Clare I work with is doing the London marathon this year.
Clare, (who is) my co-worker, is doing the London marathon this year.
Subject pronouns with “be” verbs can be deleted in non-restrictive clauses:
Clare, my co-worker, is doing the London marathon this year.