How can one atone for past sins, including an illicit relationship and actions that violate the fast in Ramadan, given the inability to fast for long, continuous periods? And does Allah accept repentance and forgive the sins for which a servant has repented and atoned?

If a servant repents with sincere and truthful repentance, his sins are erased, and he becomes as one who has committed no sin; this is based on the Prophet’s (peace be upon him) saying: "The one who repents from sin is like one who has no sin."

If semen was discharged due to your action during the day in Ramadan, your fast is invalidated, and you must make up that day. You are not required to offer expiation (kaffarah) according to the more preponderant opinion. If semen was not discharged, your fast is not invalidated by that action, because sins do not invalidate the fast according to the majority of scholars, even though intentionally committing a sin during the day in Ramadan diminishes the reward.

It is obligatory for you to make up the day if semen was discharged; otherwise, there is no make-up due upon you.