There's some assumptions being made. Psu. Can it even handle an upgraded gpu. Cooling, could it handle OC on that 4300...
The APU is a bandaid. It's a stop-gap just to get into the platform. Not a great choice for discrete gpu usage, but it is a first step.
A stronger gpu now won't do much more than improve detail levels. The old 4c FX is a serious hamper to fps limits, and depending on the op's gaming, could very easily be limiting even the rx240 fps output.
There's no easy, one thing fix, only babysteps and with a budget of $400 choices must be made on what's doable now, or risk loss of pc function entirely. OP needs a new platform. No ifs or ands or butts.
The question is $. And just how far it can be stretched. I'd not be above scouring ebay or Craigslist or other such for a used cpu or gpu or even ram.
There are options, it's a matter of how much OP is willing to invest in time and money and effort.
My i7-3770K, nzxt Kraken X61, fractal design R5 Window, Evga G2 550w, 16Gb Patriot 1866MHz, Asus gtx970, Samsung 840Pro 128Gb etc cost $600 almost 7 years ago, and $300 was just on the psu/gpu new. Everything else came to $300 from eBay. A $1600 new build for $600....
Shop around.
https://www.ebay.com/itm/124160712743
Leaves $100 ish for a mobo, plenty doable. That's not bad for ram, a gtx1060 and a R5 1600. No further need of upgrade.