"plugged in, not charging"

Lithium-Ion batteries require a specific current and voltage to properly charge (actually different levels depending on the specific charge profile: they're charged at constant current up to about 80% capacity, then charged at constant voltage the rest of the way). Charging at different voltages or with different currents might damage the battery or shorten its expected life, so the battery controller will simply not allow it and disconnect the recharge circuit.

Then, power units have a maximum wattage, and usually they have a "power curve" so that the actual voltage they give out is not always the rating voltage, but a little more (when nothing is plugged in) or a little less (if something is draining power). This is also related to the waste heat being produced by the power brick.

Finally, some laptops (and some chargers) can actually communicate and identify one another. The laptop can then actually refuse to charge, or charge more slowly, or not charge unless the laptop is powered down and the identification chip asleep, if it feels that the charger isn't its manufacturer-ordained soul-mate. If something interferes with the protocol, the charging process might be hindered. This is probably not happening to you, because HP doesn't seem to employ such strategies, but I'm including it just in case.

What seems to be happening here is that when the PC is draining current, the power brick voltage falls below what's required by the battery controller to safely charge the battery. There is no other phenomenon that can reasonably be going on between the two cases "laptop on" and "laptop off".

That said, there can be several causes for that one phenomenon. It's not even guaranteed that a single cause is responsible.

• the power brick isn't usually producing, say, "19V". Only very high-end, usually multipurpose, adapters will output a precisely tuned voltage. Usually laptop power bricks apply an AC-DC conversion method that, when supplied with AC in the 220-240 V range, outputs DC in the 18.5 - 20.2 V range. But if your wall socket voltage is actually, say, 216V, then it may well happen that the resulting DC voltage falls below minimum tolerance.

• voltage isn't the only parameter in AC-DC conversion. The phase of the current also is very important. Usually the power factor is not a concern, but if your electrical circuit involves heavy capacitive or, more likely, inductive loads (motors, etc.), the actual power going to the brick might be less than it expects. This phenomenon is used in "dimmers" that lower the power going to a lamp or appliance by counterphasing it (instead of wasting it into heat, as it would happen with a resistive load).

• the power brick might be defective. Its output (see above) is obtained by placing appropriately dimensioned capacitors and components in the circuit. If one of these fails or drifts out of spec, the power brick might start sending out less power than it should, and the battery will refuse to undercharge since this would damage the LiIon cells.

• (less likely) the battery controller might be defective. It might believe that it's not receiving enough power, even if it is, unless the brick sends more power than necessary; which it only does when the PC isn't siphoning off energy for its own purposes.

• (not very likely) the PC might be draining more than it should.

What to do

First test: get an electrical tester and verify the wall socket is supplying properly. Otherwise, check what happens moving the PC and charger somewhere else (at a friend's, at work, at a bar). The nearest to a low voltage power station, the better. Where I live you may get 238V next door to a power station, and 226V two hundred meters down the road. In some places in the country near the coast we have 218V and some appliances will complain, or not work as reliably. My own Dell M6800 charger for example takes exactly twice as long to bring my laptop from 20% to 100% charge. Mine doesn't refuse charging, but I wouldn't be surprised if some day it did.

Second test: try lowering the screen luminosity and, if possible, set the hard disk sleep time very low. Stop all applications you can and see whether you can install some CPU frequency governor to speedstep down the CPU when not in use. See what happens when the screen is dim and the hard disk stopped. Does it still refuse to charge? If so, chances are the power brick is defective (the battery governor can't be this much out of spec or your PC wouldn't even power on unless connected to the wall socket).

Third test: bring the PC to the shop and ask they loan you a power brick for five minutes. There're usually PC on show and they use the official power bricks. If their brick works and yours, plugged in the same socket, doesn't, that's quite a definitive diagnosis right there.