The credit card will have its own number which is printed on the card and should be linked to your main clubcard account (which may not have an actual card) and is usually set up when you open the card account. Have you checked your on line clubcard account ? https://secure.tesco.com/account/en-GB/manage/clubcards-on-account
or in the Clubcard app click on 3 bars top left, then Preferences and the list of cards will be at the bottom. Can't see anything in the Clubcard tab of the Tesco shopping app though.
We have 2 clubcard accounts with 3 actual clubcard numbers (but only one physical clubcard) and 5 Tesco Bank cards each with their own clubcard numbers but all link to one joint account to accumulate the points and our coupons have Mr & Mrs on them.