The part of the sentence containing this phrase is "to alleviate the many ills to which the flesh is heir".
The question is "Could you paraphrase this correctly?"
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"to alleviate the many ills to which the flesh is heir" = "to alleviate all the bad things that humans suffer.
The expression to which the flesh is heir is taken from Shakespeare's play Hamlet (Act 3 Scene 1).
To die, to sleep—
No more—and by a sleep to say we end
The heartache and the thousand natural shocks
That flesh is heir to—’tis a consummation
Devoutly to be wished!
To which flesh is heir or That flesh is heir to is a poetic way of saying "that afflict us" (literally "that our bodies inherit")
Flesh means "mortal body", and is as opposed to the spirit or soul.
"Flesh" is the muscle, fat and sinew of the human body, what we might call "meat" in a non-human animal. By synecdoche (using a part to refer to the whole), it means "body" or "physical form".
"To be heir to something" (in this case, "which", referring to "the many ills") means "to inherit something" or "to get something by means of a will". Metaphorically, the "will" (as in "last will and testament") is dropped and "inherit" instead implies a certain inevitability to the "getting". If you really wanted to push it, I'm sure you could come up with something about God's will.
In short, "the many ills to which the flesh is heir" refers to the many diseases, injuries, and other problems the body gets because it's a body with a physical presence in the physical world. This isn't just age or fatigue and their effects, by the way. It also includes actual diseases and injuries, as opposed to more spiritual or emotional ills, such as greed or heartbreak.
"To alleviate", meanwhile, means "to soothe" or "to lighten" (as a load).
So, to paraphrase,
"to alleviate the many ills to which the flesh is heir"
"to soothe the many problems the body gets because it is physical"