Yes, that is a pretty weak dual-core CPU you have there. It's an 8th-tier CPU (http://www.tomshardware.com/reviews/cpu-hierarchy,4312.html), & being a Socket AM1/FS1b CPU it's going to be hard to find a better one. In fact, of the supported CPUs (https://www.gigabyte.com/Motherboard/GA-AM1M-S2H-rev-10#support-cpu), I couldn't find any of the other ones (including the quad-core Sempron) on the hierarchy list. Nor could I find information (even on AMD's own site) for those "Athlon X4" CPU models listed.
However...
I did find some info on the "Athlon 5370" APU, which appears to be the fastest CPU you have available. Based on AMD's specs for it (http://products.amd.com/en-us/search/APU/AMD-Athlon%E2%84%A2-APUs/AMD-Athlon%E2%84%A2-Quad-Core-APU/Athlon%E2%84%A2-5370-APU-with-Radeon%E2%84%A2-R3-Series/182), it looks like it's extremely similar to their A6-5200 APU (http://products.amd.com/en-us/search/APU/AMD-A-Series-Processors/AMD-A6-Series-APU-for-Desktops/A6-5200-with-Radeon%E2%84%A2-HD-8400/41). Neither one, unfortunately, is on the hierarchy list. If I had to guess, however, I'd say it's roughly comparable to the A6-3670K, which would make it a 5th-tier chip.
The benefit you'd get from the CPU is more cores (going from a 2C/2T CPU to a 4C/4T CPU) & much faster core speeds (about 50% faster). It's in no ways going to be anywhere near any kind of "gaming" machine, but it would probably help out for normal workloads. The only issue is going to be finding one available (used or new).
Barring that, you're limited because of using 32-bit Windows on your RAM. What I would strongly recommend is a couple of things:
1. Get a discrete GPU. Doesn't have to be anything powerful -- in fact, a cheap GT 1030 would be more than sufficient. The main benefit is that, instead of having some of your system RAM being used by the integrated graphics, it can be handled by a separate GPU.
2. Get an SSD, at least 120GB but preferably 250GB, & reinstall your OS there. Again, there's not a whole lot of benefit for gaming, but your startup/application loading times will drop significantly. The only caveat is that, with only 2 SATA III ports on that motherboard (& the assumption being that they're already in use because of your existing HDD & a CD/DVD drive), you'll either need to switch over to an external CD/DVD drive, find an external USB enclosure for your WD Blue drive (to use for secondary storage), or find a SATA controller card to install in 1 of the PCIe x1 slots to make room for the drive. Plus, I don't know how much room there is in your case.
3. Consider reinstalling Windows from scratch & converting to a 64-bit version. Even though you only have 2 DIMM slots, they can each handle up to 16GB DDR3-1600 sticks (https://www.gigabyte.com/Motherboard/GA-AM1M-S2H-rev-10#sp). Switching over to 64-bit Windows would allow you to go up to 8GB or even 16GB of RAM, which would help speed up your system.