No problem helping out.
If your Linksys is old.. ie G wireless it is better to replace it.. A brand new TP-Link model like WR841HP is perhaps $40. The next model up.. WR1043ND should be around $60 and is faster chipset with gigabit. Wireless will be identical. Then AC model like C5 or C7. It is up to you what you want to spend.
There is no problem using a mixed wireless network.. I found it much better to do exactly that.
Apple router for the apples.. (Banana router for the Bananas 😁)
But you can start today with the old Linksys so you can get the HP on the network.
(Well you can do that already by ethernet of course) But if you want to use wireless.
Most of the linksys routers will not have an AP mode.. (that means just using router as access point). But all wireless routers can be used in a method called WAN BYPASS.
1. Plug the Linksys into the HP by ethernet.
2. Open the browser and go to the 192.168.1.1 (or whatever the IP.. but usually that one).
3. Change the IP address to the same range as the AE. Open your airport utility on a Mac to find this.
What you see here, will depend on the network setup.. If it is router on default it is easy.. The AE will be 10.0.1.1 but if it is bridged as mine is (almost) you will see any old strange IP.. take note of it.
We need to give the Linksys a fixed address. In my case I would pick 192.168.2.100 (making sure nothing else has that IP).
If your AE is default 10.0.1.1 then I would set linksys to 10.0.1.220.. or anything above 200 up to 254 will work.
Write this on a label and place it under the router.. along with username and password.. you will find you have to have the address in the future.
If you need help with this tell me the whole network layout. modem and router setup.
4. Setup with wireless in the Linksys as you like, with a different name from the AE.
5. Most important.. now turn off DHCP server. And actually this might be better done before step 3.
6. Plug LAN port of the Linksys to the LAN port of the AE.
It should now provide wireless to your HP but it is fully integrated into your network and you can communicate across all the clients as well as internet.
It is easier to double NAT.. but this fixes it so you have full network connectivity as well as internet.