If there is no question of the woman even partially supporting the man, then it’s just a lover. “She’s always had lovers, I knew about them but didn’t care.” That sort of thing.
If an older woman is at least partially supporting a younger man, he might euphemistically be a “protégé,” especially if he has pretensions in the arts (c.f. Carlo in My Man Godfrey). It doesn’t have to be sexual, it may be just aspirational.
The pool guy?
I’m surprised the term “back-door man” isn’t better known than it seems to be from reading the posts here. Besides the Howlin’ Wolf song (written by Willie Dixon and later covered by the Doors), the term appears in other song lyrics, notably Led Zeppelin’s Whole Lotta Love. (the link is teed up to the lyrics “Shake for me girl, I wanna be your backdoor man”).
I thought it was Boy-toy.
Someone else said friends with benefits.
I was sure that meant the benefits were of a sexual nature. 
I’m also surprised that “back door man” isn’t universally known. I was introduced to it by Whole Lotta Love in my youth, but I thought it was well known as a term for a man who sneaks in the back door while the woman’s husband is away.
It also has another meaning, a man who practices anal intercourse, and even (according to the Historical Dictionary of American Slang), a proctologist, but I think those meanings are used less often.
I heard the phrase in a few blues songs. But I thought it meant something different.
While lessened stigmata may be more recent, given class was a thing and given human nature, I would guess the use of a specific word was rarer than the need.
For lots of us, songs are not a place to gather msinstream vocabulary.
I don’t recognize any of the mentioned song titles. Nor most of the mentioned artists. I might recognize some of the tunes if I played them.
It’s not that the backdoor man expression isn’t known, I just never hear it used.
Or, to use a term I saw used in Hong Kong some years ago: Weekend husband.
A goombah or escort.
I would understand “back door man” if it was used, but it strikes me as a bit too poetic for everyday use. “He’s her back door man” isn’t something that you’d hear used in everyday conversation.
Now “He’s her side piece” fits the bill nicely, and is a phrase I’ve actually heard in a conversation.
What about “mister”?
This is the kind of euphemism that I despise. It obscures instances of actual long-term companionship or friendship for the sake of trying to disguise an illicit lover’s relationship. It makes things less clear.
“Woman of a certain age” is another one. What age? I’ve never even been sure what the expression is trying to get across.
I have never heard it outside of this song. I had a more sexually explicit interpretation of that phrase 
Italian cicisbeo is pretty much it, unless the financial-support angle must be implicit in the word.
Cicisbeo is supposedly borrowed into English, but its use seems to be exceedingly rare.
You do not recognize The Doors or Led Zeppelin? Heck, my 22 year-old nephew knows them.
To be fair though, I never paid very close attention to the lyrics enough that I wondered what a “backdoor man” was. Most rock lyrics don’t make much sense to me so I tend to give them little attention (usually).
I’ve only seen it in Regency era romances. It’s used about a guy who escorts a married or widowed woman to social events, where the husband, if alive, is not interested in that type of event, but there is a strong implication that the cicisbeo performs other husbandly duties. The monetary aspect is mostly absent, except for the woman possibly paying for minutiae.
What does that mean? (I Googled it and it was still unclear…kinda sorta got a sense of it but maybe not.)
It shows up in widely recognized songs, but the usage of “back door man” outside of those lyrics is pretty much confined to early 20th Century Southern Black culture.
Boytoy is the one I’ve heard the most, although it almost always applies to a younger man, “kept” by a successful older woman, who may or may not be married. IE it isn’t exclusive to a married woman’s lover. And isn’t always full financial support in the way mistress often implies.
I have seen kept man used, with more or less the specific meaning of a fully supported lover, who was expected to be exclusive to the woman, who didn’t have to be exclusive at all. And yeah, it overlapped with the trope-ish young, tan, blonde pool boy that lived on site. 