I don't think that changes anything.
4060 is the fastest and most expensive thing you should buy, without overly-stressing your PSU.
1060 6GB for €80, used, is probably the most cost-effective thing you can buy, without overly-stressing your PSU.
Realistically, if you don't want to change your PSU, those two are pretty safe bets, and of those two I wouldn't bother spending the €250 more on the 4060 unless you have plans to upgrade the rest of the PC in the next year or two, because your CPU will hold it back a lot.
Both of those are still a gamble; The 4060 might give you UEFI-compatibility issues with your old motherboard, requiring a BIOS update that may not even exist. I *think* the P8H77 is going to be okay, and I don't see any BIOS updates for it that mention UEFI GPU compatibility, so hopefully it was included out of the box with support. Ignoring the compatibility risk, both the 4060 and 1060 are 120-Watt cards which is 40W more than your existing Radeon. If (and this is only an 'if') your PSU is on the way out, that extra 40W of power draw might be the final nail in the coffin. I personally think you'll be fine - most old systems I see with RX 480, RX 570, GTX 1060 cards tend to have equally old 450-650W PSUs, some good quality, some not.
Your PSU is an unknown, so literally anything you do that increases the power draw is a gamble. If you don't want to take that gamble, get a GTX 1650 or 3050 6GB, both of which use less power than your existing Radeon - but they're not exactly great value cards.