2 tenants applying but only want to take out tenancy in one persons name only

Tenancy Types and Management

I’ve had a couple of tenancy applications recently each involving a couple where they’ve requested the tenancy be taken out in one of their names only. It transpires that in each case, the person they want to leave off the tenancy has poor credit history which explains a lot. The other applicant on each respective application has a satisfactory credit history and salary which easily meets the affordability criteria.

I’m personally not comfortable with only one applicant signing the tenancy for the following reasons:
1) If the couple split up and the person who’s on the tenancy decides to leave thereby leaving the unnamed person on the tenancy in the property, what legal recourse would I have to evict if she couldn’t pay the rent, particularly as their credit history is poor and she isn’t even mentioned on the agreement?
2) Even though the named applicant may have a good credit history and can afford the rent, his partner may have access to his funds (e.g. joint account) thereby presenting a risk of non-payment of rent.

Maybe I’m being overcautious but would welcome any thoughts from other members.