I've been going once weekly to a hagwon, and there I do 20-minute classes with up to four students at a time. I've gotten a lot of mileage out of bomb games and card games like Uno and such.

Now I'm running out of ideas. Bomb games have gotten tedious, and I think a lot of the students are tired of them too. Uno and games like Old Maid and Go Fish are fun, and everyone likes them (including me), but they aren't really educational. I should avoid them.

So what else? What can I do in 20 minutes that is fun and educational (or at least gets them to use some English)?

There are some variables that limit my choices, I think. For example:

-Sometimes there are four students, sometimes less. Sometimes even just one.

-Their ages run from 3rd grade elementary to 1st year middle school. Their skill levels, of course, are also greatly varied. And they're put into classes often higgledy-piggledy based on whoever is at the hagwon at the time and is free to have a class.

-It's a reading hagwon, but the owner doesn't want me to focus on that. Speaking and such is better. No worksheets, no textbooks, etc.

So... Yeah. I'm not crazy about doing chants or watching videos because, as I said, some are older and resist sing-alongs (and it's just awkward for me too, trying to get a 13-year-old excited about something that also interests an 8-year-old).

Any ideas? Other games or setups to help pass 20 minutes?

you could just treat these as shortened versions of regular classes.

5 minute warm-up, 5 minute vocab/dialog practice, then 1x10 minute speaking activity.

having the differing age groups could be challenging, but you could split them into 2 groups. older students in one group, younger students in the other. then you can use the same activity, but just with different dialogs.

it might sound like a lot more work but it doesn't necessarily have to be. for example, maybe the sentence for the younger students is "I like baseball." for the older students, it could be "I like baseball. You play it with a bat and a ball. You hit the ball with the bat" or something like that.

There are plenty of board games on this site (Shoots and Ladders style). U could edit a template and have two different levels. High level kids can practice harder words/sentences. Lower level can practice easy words/sentences. U can practice all the target language together as a class first so youre not really separating.
There are also the "flick games" that u can edit for your target language.
There are various versions of Battleships on this site.
For all of these, even if one student comes u can play against them.
U can do easy versions of Pictionary or Charades for some target language.
If you have more than 6 kids u can play Heads Down, Thumbs Up (aka Heads Up 7 Up). There are a lot of ways to incorporate English expressions.
U can do 4 corners, or if the room is small just have students/teams choose a number and write it on a white board/use their fingers.
U can play line bingo which is a little more challenging because they can only "mark" a word/phrase if its at the end of their strip.
There are a bunch of templates for "whats behind" or "elimination" games. Similarly there at least two versions for "word lotto."
There are many templates for slap games (ppt versions u play as a group or doc versions u print and they play in pairs)
There are a bunch of other card games people have made and posted here but I cant remember them all by name. Look around or maybe do a google search for 'card games.'