"Do you know any method of making a second recovery partition?"
Basically you are down to the methods already suggested. There is no way that I know of to now re-create the original & now lost recovery partition that would work. You wouldn't be able to again create recovery media like you previously did anyway.
Really your best options are download Win 10 & do a clean install, or purchase the recovery discs from HP, which after a year or two, those would then restore a dated version of Win10, & you would be looking at more work updating to the [at that point in time] current version of Windows.
As it now stands MS is expected to release a major update possibly around October this year. When that happens I will download & burn the new version to a DVD, adding to my library of windows so to speak.
If you do clean install & get your system setup with all updates, software etc, you could then either Clone or Image the hard drive, saving such to an external device. There are numerous software programs available for that purpose. Many are free.
A list here: https://www.provendatarecovery.com/blog/10-free-disk-imaging-software-tools-for-harddrive/
Of the 10 listed, the most popular are AOMEI Backupper, Macrium Reflect, & Paragon Backup & Recovery. You could be back up & running in short order. Periodically, due to Windows Update, new installs/uninstalls of software etc you could create new backups & then have more than 1 to select from.
Both cloning & imaging create an exact record of your drive. That includes all the files on the drives along with the master boot record & everything else your PC needs to boot.
Disk Imaging: Imaging creates a large compressed file of your drive. Then you can restore that file to bring your drive back to life. Because the image file itself is large, users will save them to external drives.
Disk Cloning: Cloning creates an exact, uncompressed replica of your drive. If your hard drive fails, you could remove it & replace it with the cloned drive. Cloning can get you up & running quickly, but it doesn’t offer as much flexibility as imaging. I’d recommend an image over cloning.
Do you own an external HD? If not, HDs are cheap these days & everyone should have at least one.