So in my setup at my home for example, I use one router as the central hub. It serves out DHCP to my home network and handles all of that. I run another router in my upstairs used as an access point to simply extend my wifi network. So in effect, in my upstairs, there is a second wifi network that my wife in her home office connects to, but all of the network addresses etc all come from the router downstairs.
To get to that point, i ran an ethernet cable from my downstairs to my second floor where the secondary router sits. My access point is just an old router I had so I hooked it up, and disabled DHCP on it, so basically it serves the same function as a simple access point. It's not doing routing or anything, just putting out a signal.
In your case, you said you had an access point that was apparently the only way the PS4 for example will connect but it was having issues etc. I don't know if that was connected via wireless or by a wire, but with those powerline adapters, they are supposed to make the unused electrical wiring act like ethernet cable. So what I was meaning is if you are going to drop an access point for the PS4, then you would probably want to run from the main router, into the powerline adapter, then when it pops out the other side on the powerline adapter, take that ethernet wire into your access point so that you know it has a hardwire connection and not relying on a wireless signal which can be flaky.
Now as far as the kids getting high pings etc, if their connection for that comes in like this main router->powerline->client
Then you need to see first of all if you can plug a system maybe a laptop or whatever into their modem or device that comes from the ISP to see what type of signal you are actually getting from the ISP. So in other words you want to troubleshoot this way.
1. Plug system into ISP modem and go to a website say speedtest.net and run a test. Maybe someone else can recommend another site for checking ping
2. If all looks good at the ISP modem, plug a wire into their main router, repeat, make sure you are getting a good speed and ping etc. Do this on wired connection and on wireless
3. If the main router looks good, then work back and test each of the pc's plugged into the powerline adapters. If those test out good with your system, but not theirs, that person may have something wrong with their pc. If their pc tests out good, maybe the server they play on just has an issue.
4. Test the access point as well.
In other words, just start where the ISP brings in a connection, and work your way 1 device at a time backwards. For a home network they've got a few moving parts, so you will need to isolate it one part at a time. For example if you get good results from the ISP's modem but then plug into their router and have a bad connection, you may need to replace the router or update it's firmware etc. If their router works fine but coming from the powerline adapters does not work well, you may have an issue with one of the powerline adapters. When you test the powerline adapters, since it sounds like there are 2 in the mix here, disconnect one completely while you test the other one, just in case they are interfering with each other. Never really used powerline adapters, but since it does use unused frequencies in electrical wiring etc, it stands to reason that the two cold interfere with one another.
Depending on everything you run into, instead of using the powerline adapters, it may be preferable to run an ethernet cable to a central location between where the kids rooms are at, and see if you can drop a strong access point there, and get them both a good wireless card for their systems so they can connect via wifi if the powerline adapters are flaking out. Of course you could always try different powerline adapters

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I will say though, if they are using an ISP provided router for their wifi/routing, that could be a major part of issues. I don't tend to trust routers provided by the ISP. Their quality and speed does not always seem great. I once had ATT dsl, paid for 12mbps. I was getting 10 or so of that at first, so ok I can deal with it. Started getting signal dropouts and bad throughput, got irritated with it(the unit was one of those modem/router all in ones). So I got annoyed and disabled the wifi on their unit and set it to where it was only allowed to give out maybe like 10 IP addresses, and set up my own router to come out of that and told it to handle all the routing etc. Magically my speeds, ping etc all came back where they should have been. Didn't have a lot of issues after that.