Keeping only your spouse’s surname
HM Passport Office’s guidance covers this exactly. If it issued you a passport in a double-barrelled name made from your previous surname and your spouse’s, and you now want only your spouse’s surname, it says it can’t use the marriage or civil partnership certificate. You need evidence of the change of name and evidence that you use the name for all official purposes.
Your marriage certificate shows how you got to Clark-Price, but not how you got from there to Price. A deed poll fills in that step: it records your old name, your new name and your declaration that you’ve adopted the new one.