Can I install a second hard drive in my AcerTC-875- UR12

@sawfish

Assuming that you are now using one spinning HDD  with the Windows 10 on it and an optical drive is being used too, you should be able to add another 2.5" SATA3 SSD with the still open SATA port on the motherboard.

If the motherboard still has two SATA3 ports open, then it means you are now using a M.2 SSD (probably a PCIe 3.0 x2 M.2 NVMe SSD originally shipped) as the boot drive. 

However, just like @Billsey just mentioned there is a M.2 slot, it should be used as the boot drive with the Win 10 on it. The reason: PCIe 3.0 x2 M.2 NVMe SSD is 2-3 times faster than regular 2.5" SATA3 SSD. If you use a PCIE 3.0 x4 M.2 NVMe SSD, it will be like 5-6 times faster. The M.2 NVMe SSD looks like a half size memory stick, unlike the 2.5" SATA SSD that looks like a small box.

This M.2 slot is usually partially blocked from view by the optical drive tray. To get full access to this M.2 slot, you may need to remove the optical drive first.

@iamthewalrus
One thing I like to point out is that the designed hard disk tray can only keep one 3.5" HDD( unless you want to drill holes yourself). As seen from the photo given by @StevenGen, only the four orange color holes are for that one 3.5" HDD. One 3.5" plus one 2.5" HDD/SSD or two 2.5" HDD/SSD can by used there.

 For @sawfish and @iamthewalrus ,   My personal experience is:

I have one PCIe 3.0 x4 M.2 NVMe SSD in the M.2 slot as the primary drive with Win 10 on it

A 3.5" old fashion spinning HDD mounted to the four orange holes of the HDD tray, just use it for Win 10 system image backup. I connect it to the PC once every three or four months, as there is no need to do backup everyday or every month.

A 2.5" SATA3 SSD with a Linux version placed on the tray at back side of the optical drive (there is a tray with no holes drilled, good enough to put the 2.5" SSD using Gorilla tape). The power connector and the data cable will be shared with the 3.5" drive in case I need to do the image backup.

A 2.5" 10,000 RPM sever drive with data only always connected to the PC. This one shares the data between Win 10 and Linux . This is mounted to the HDD tray.

Other than that, I have another x4 M.2 NVMe SSD added to the PCIe x16 slot through an adapter. Currently having another Linux on it ( because I have no intention to use the PCIe x16 slot for a graphics card). 

This kind of adapter can even support one M.2 NVMe SSD plus a M.2 SATA SSD on the same card.

NOTE: I always suggest people to use x4 M.2 NVMe SSD if the PC hardware is capable of that. As it is the fastest drive.

If not, try to use 2.5" SATA3 SSD, as a second choice, as it is still a few times faster than old spinning HDD.

I've found the last usage of my 3.5" spinning HDD is using it for image backup instead of throwing it to the trash can. (Don't worry, I have other backup  in addition to this old HDD).