What size is your HDD? How many partitions, of what size, particularly the one containing the C drive?
What OS is on the HDD?
If I do a clean install of Windows 10 on the SSD, I assume I will have to reinstall all the programs that were installed through Windows in the past onto the HDD.
Yes, that's correct.
Do you still have the installation media / licence keys for the applications you want to keep?
I would think about cloning the existing C drive to the SSD (assuming that the original OS is what you want), provided that the C drive partition is the same size or smaller than the SSD.
It's possible to use disk tools to split and / or resize existing partitions, but it might be simpler to just buy a bigger SSD - storage is so cheap nowadays, about £50 for a 1TB SSD.