While it's got little to do with the original hunter issue...

Summons/Rituals are pretty counterintuitive. There's more than half a dozen rites in game, each requiring a different thing to be sacrificed. Or not.

To summon something, you need a knock component. Either some knock lore, an item with knock resonance, or a cultist with knock levels.

You also need 2 more influences, depending on what you want to summon. The maid-in-the-mirror for example, a rather beastly creature to summon, requires both winter and edge influence. Some summons seem to require a corpse as well, but I've had little luck with those so far.
So, you need to fill your rite slots with things that provide those levels. Depending on the rite, you might have to sacrifice lore, an item, an influence (which are powerful, but volatile as hell, sticking around for a VERY short time), a follower.

Some of the lore cards provide hints, but there's more than one way to summon a Byakhee, so to speak. The exact combo matters little, as long as you provide enough influence levels.

Summons have a limited lifespan, but some can be "recycled" rather painlessly by sacrificing them in order to summon the exact same type of creature. They usually provide the influences required to summon them by themselves, so all you need is a bit of knock.

For shit and (insane) giggles, you can also portrait them as a painter.

There are also some Rituals that allow you to shift reason to passion and vice versa, and restore health from decrepitude. They only seem to require 2 influences, but very high levels of them in order to succeed.

And of course, there is that pesky ascension business at the end of the game, which requires a rite and very strong influences as well. Unless you come across that one rite that allows you to sacrifice half a dozen things at once, using a summon to provide the needed influence makes that whole ascension thing a lot easier.