A man with a wife is a husband, a man with a concubine is what?

Solution 1:

Master:

• the man in authority, such as the head of a household, the employer of servants, or the owner of slaves oR animals.

Concubine:

• a woman who cohabits with a man to whom she is not married, esp. one regarded as socially or sexually subservient; mistress.

From The Designation of the Individual: Expressive Usage in Biblical Narrative

... her husband is "master" from the viewpoint of status. In this sense, "concubine" contrasts with "woman" in the sense of "wife". However, from the point of view of personal relationship, "master" and "concubine" are "husband" and "wife".

Ngram: master and concubine

Solution 2:

There are several quotes that describe a man keeping a mistress, as her protector.

Did a mistress wait for her protector to come to her, or did ...

"Unmasking the Duke's Mistress" By Margaret McPhee

A mistress should never be foolish enough to fall in love with her protector, nor trust him ...

"For Desire Alone" (Mistress Matchmaker #2) by Jess Michaels (Goodreads Author)

Solution 3:

You might consider,

concubinator

Rare A man that keeps a concubine.

OED

concubinate

n. Concubinage

Webster's Revised Unabridged Dictionary

In some cases, concubinage is monogamous; that is, concubine and concubinator live together in a quasi-marriage [...]

The Oxford Encyclopedia of Women

paramour

A person with whom someone is having a romantic or sexual relationship and especially a secret or improper relationship.

M-W

It is well settled, by the decisions cited in the prevailing opinion in this case, that a paramour and concubine who succeed in concealing their cohabiting as man and wife are not guilty of living in open concubinage. Southern Reporter

...and dark cull(y)

A married man with a secret mistress.

A Dictionary of Slang and Colloquial English

A married man that keeps a mistress, whom he visits only at night, for fear of discovery.

Grose 1811 Dictionary

Etymology

cully n.1 (also cullee, culley) [? as ‘fool’ there may be links to Ital. coglione, a dolt, but as ‘man’ it may well come from the Sp. Gypsy chulai or Turkish Gypsy khulai, both meaning man, or poss. fig. use of French couillon, testicles]

(Oxford Reference)

Solution 4:

I would suggest the sometimes derogatory but modern-day term sugar daddy

Sugar daddy is a slang term for a man who offers support (typically financial and material) to a younger companion, e.g. a kept woman
Wikipedia