Not sure about Hearts of Iron or World of Warships. Civ 6 is pretty CPU-intensive; the graphics will be fine, but you might see some issues with the time it takes the AI to complete its turns.
That being said...in terms of "integrated" graphics, these must be the ones built into the motherboard (not the CPU), as that Athlon is a chip harvest of the Provost/Deneb Phenom IIs (which didn't have integrated graphics). That being the case, do you happen to have your motherboard model? Depending on the exact motherboard, you might be able to upgrade your CPU to a full Phenom II CPU -- something like the X4 920 to 945 or X6 1045T/1055T models at a minimum, but you might be lucky enough to go up to the X4 975/980 or even X6 1090T/1100T. You might be really lucky enough that it's a Socket AM3/AM3+ motherboard that can handle the later FX chips, like an FX-8300/8320/8350/8370. Again, not the best CPUs to buy for a brand-new system, but a fairly inexpensive upgrade (maybe $100-200USD tops) for your existing system that at most should only require a BIOS update (i.e. no need to completely reinstall Windows or buy any other new hardware).
While you're at it, are you using a 32-bit version of Windows or 64-bit? If 32-bit, make sure you've got the full 4GB of RAM that it will take (preferably as two 2GB sticks for dual-channel mode). If 64-bit, see if your motherboard can handle 8 or 16 GB of RAM (again, buy a paired kit so you can use dual-channel mode).