@A myHIVteam Member, maybe in Uganda they do not have proper medication to treat syphilis, but here in the United States, syphilis is a 100% curable infection. Or you are thinking about herpes, which is not curable, but manageable.
This is the cure for syphilis in the United States:

https://www.mayoclinic.org/diseases-conditions/....

Syphilis - Diagnosis and treatment - Mayo Clinic

https://www.mayoclinic.org/diseases-conditions/syphilis/diagnosis-treatment/drc-20351762

Syphillis is a bacterial disease, it would only come back if you got re infected or if treatment was sub optimal. You will show a syphillis antibody positive for life once infected , so you should let any new sti clinic know that you've been previously treated, so that they can do a secondary blood test to confirm the result.

@A myHIVteam Member, there are cures for many things, but reoccurrence can still happen. For instance, the plague, covid, chicken pox, measles, and more. Even being HIV undetectable, you can be reinfected to be detectable.

Yes. Penicillin by Injections. If you are allergic to Penicilllin, the drug known as Doxycycline capdules can be used twice daily for two weeks. Doses must not be missed if using Doxycycline for Syphillis and you have to avoid direct Sun if using Doxycycline.

Yes, syphilis is treatable and curable in people with HIV, but HIV can make it harder to treat, especially if the HIV is advanced or untreated. The recommended treatment for primary and secondary syphilis in people with HIV is a single 2.4 million unit intramuscular injection of benzathine penicillin G. Additional doses of penicillin G, amoxicillin, or other antibiotics are not more effective.