Without actually looking at the eBay item, the only things I've really used PoE for is external wireless access point hardware. Only a single Ethernet cable needs to be sent through from the outside, in a suitably weatherproof/wildlifeproof way according to your own local expected conditions, and once safely inside this plugs into a PoE adapter which is like a male/male RJ45 connector with a power input (usually a mains plug/transformer combo leading to a coax power plug).
The PoE, in this case, doesn't do anything 'to' the end device (nor, assuming you've got the adapter plugged the right way round between cables, the computer or other hardware you have at the other end). If they're using PoE to reduce the cable clutter (or at least set it at the computer desk, where you may have loads of sockets, rather than where the
POTS connection is, which may not be so blessed), than you've still got a router. (Or at least a single-Ethernet-out modem, although I don't see many of those[1].)
And all that's assuming that your "Power over Ethernet" isn't a typo for "Ethernet over Power", which is a different beast altogether...
The standard first question I'd be asking if I knew I was looking at a British internet connection that was playing up is... Are you using microfilters? You say DSL, of course, rather than ADSL, so I'm not sure exactly if you're set up with same sort of system, but hereabouts ADSL modem/routers plug through a microfilter adapter/thingummy to avoid non-data carrier signals interfering with the modem's reception (and the data-carrier signal leaking onto the voice-phone network). This can cause problems with data-rate. Every standard phone needs to be plugged into the standard phone-type socket ('BT' plug, in our jargon... this is doubtless different for you) through the phone-style side of a microfilter and the (single![2]) router is connected via an RJ11[3] into one of the microfilters (whether shared with a phone or not).
Above and beyond that, some home phone extensions have been added beyond the ability to deal with electrical noise adequately. Or, unfortunately, pass by some noisy power coupling, somewhere along the way. That latter is harder to work out, but connecting the router (or whatever it is) to different bits of your internal phone system might give you less a noisy extension. (Really, it should be connected at the "master" socket, or as close to the point the line comes in from outside as possible, but I've no idea if the same sort of system is applicable in your part of the world as mine, so I'll leave that for you to work out.)
I'm sure someone will point out some flaw in my hasty response, though, especially with any inherent geopolitical differences that I'm not taking account of.
[1] Almost universally, these days,
British ISPs seem to send out wireless-enabled routers with up to four Ethernet jacks. In my experience. YMMV.
[2] There was that time someone had plugged
two routers into his single (but split to different extensions) phone line... And wondered why only one would work at a time, and sometimes neither did...
[3] Is that the standard US phone cable end? It seems to be so universal, I assume it's a standard from
somewhere, just needing different adapters to plug into various other countries' phone-sockets.