If clicking a field doesn't turn it editable, check these in order:
Is the profile locked? Non-admins can't edit any field on a locked profile, look for the Locked status banner.
Do you have edit scope? View-only users can see fields but can't edit them.
Is the candidate outside your ownership boundary? Some edit scopes are restricted to owned records only.
Is this a primary email or phone? Primary contact entries can't be archived or deleted directly, you have to designate a different entry as primary first.