My partner and I have been together 17 years and we are wanting to bring another male in to our relationship (MMF). What would the correct name be for this relationship? Both males are straight.
Well, for one, I can't quite parse exactly what kind of 'relationship' you're trying to name. If both males are straight, then they probably won't want to be in a sexual or romantic relationship with each other, so the new person probably won't be "brought into" your existing relationship - they'd primarily be with the female partner, right? Dating one person in a relationship is different than "joining" or "being brought into" a two-person relationship. You'd be well served to think a bit more about the specific dynamic you're hoping for, so you can better explain it to yourselves and future partners.
Are you planning for this to be primarily sexual, like someone to have threesomes with? The word for that is really just "threesome" - or "group sex," "multiple partners," that sort of thing. There are specific sexual concepts common in MMF sexuality, like "cuckolding." Are you looking for someone for the female partner to date? That is just called "polyamory," "V-shaped polyamory," or an "open relationship." One person sought after by an existing couple to date that couple as a unit is called a "unicorn."
There are not, to my knowledge, specific terms for relationship configurations based on the specific number and genders of people involved. I'm not sure we need those, frankly. If you are looking for linguistic validation that what you want is "a real thing," that also doesn't exist, and you don't need it. If you are looking for a very particular term that you think will perfectly describe what you want so that you don't have to do other work of explaining, defining, and communicating, that doesn't exist. You need to continue to do the work of identifying and defining your needs and boundaries.