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You could if you wanted to, sure. You have an AM4 socket, the most versatile CPU socket made in the last 20 years. You could put ANY Ryzen 5000-series CPU into it (make sure the BIOS has the correct upgraded version of course) from the R5-5600 to the R9-5950X and anything in between.
The thing is, the RX 6400 isn't powerful enough to bottleneck the R5-3400G so upgrading your CPU would make no difference. You'd need at least an RX 6600 to be able to properly utilize a Ryzen 5600 CPU. Now, you CAN get that card because according to the TechPowerUp GPU database, AMD recommends a 300W PSU. The only question is whether or not you have a connector for it because it needs a single 8-pin but it draws less than 150W so one of these would work just fine if you need it:
MOLEX to PCI-Express 8-pin Power Adapter
The RX 6600 isn't expensive either, there's a reason why it's considered to be the best value video card out there right now.
ASRock Challenger D Radeon RX 6600 8GB: $235 at Newegg
According to the TechPowerUp GPU Database, the RX 6600 is 130% faster than the RX 6400 while only being 68% more expensive. That's right, it's over DOUBLE the speed of the RX 6400 and I think that your PSU can handle it. Techspot's review showed that their test system pulled a maximum of 297W when running Doom at Nightmare settings at 1440p. If you only game at 1080p, you wouldn't even come close to that number.
Once you have that in your PC, then you can look at upgrading your CPU or PSU because spending money on an RX 6400 when you want to upgrade other things would be a bit of a waste. I was under the impression that you were satisfied with IGP performance and didn't want to bother upgrading anything else. If you DO want to keep upgrading, then the first step would be the RX 6600. It's only $35 more than the R7-5700G and you wouldn't have to buy a new motherboard or RAM. Your 16GB of RAM is just fine and it'll be at least a year or two before you have any real issues with that.
The RX 6600 with your R5-3400G would give you gaming performance so good that it would be like night and day. You'd be in absolute gaming heaven, especially if you're used to using an IGP. Since you'll still be using an AM4 motherboard, you will always be able to upgrade to an R5-5600 or better if you want to and you'll be able to grab a used one for cheap from eBay. Before doing that however, I'd grab a new PSU because you'll already be VERY happy with the performance that your PC is putting out even with the APU that you already have. I think that a 500W 80+Bronze or better would be a good choice for current hardware and a look to the future, but for about the same price as one of those, I found a 600W 80+Gold-Certified unit:
Thermaltake ToughPower 600W 80+Gold: $60
You won't find a better PSU deal than that. That will give you a strong foundation to work with (because the motherboard and PSU are the foundations of any PC) at the R5-5600 CPU if you want to (or you might just want to stick with what you have). Either way, this PSU will make your platform very upgradable and it has a 5-year warranty. With it, you'll never have to worry about what video card or CPU you decide to use (within reason of course) as even the RX 6700 XT only recommends a 550W PSU and that's three performance levels above the RX 6600.
So, the total amount that you'd need to have a stable and capable gaming PC with no concerns at all would be $295 and you don't need to get the PSU right away because I think that the one you have would be fine for quite awhile with an RX 6600.
As for the R5-5600, right now, it costs
$150 from Newegg and it will be cheaper than that by the time you decide to get one. In any case, that's the best way to spend the money that you have because you'll be getting the best value from your GPU by buying the best value GPU, you'll be getting the best value PSU and you'll be able to get the best-value gaming CPU.
In any case, with just the RX 6600, your PC will already be more potent in gaming than any console ever sold. We're talking over 60FPS at 1080p in Assassin's Creed: Odyssey, over 100FPS at 1080p in Battlefield V and over 80FPS at 1080p in Fortnite. This is without any CPU upgrade, this is JUST with the RX 6600 for $235 and believe it or not, your APU is the bottleneck as if you ever want better than this, upgrading your CPU will allow the RX 6600 to go faster than this. Having said that, if you have a 1080p60Hz display (like most people do), you won't need to replace your APU anytime soon.