And still with sicknesses and shameThou didst so punish sin (12-13)
Fragile and sad to begin with, the speaker sets off on a life of strength-sapping illness and embarrassment, all meant to punish the sins he inherited and the sins he just felt like doing himself.
That I becameMost thin (14-15)
"Thin" here definitely has the usual meaning of really skinny, but in the context of the speaker's life of sin and sorrow, it also means spiritually skinny. The speaker isn't feeling close to God and his religious strength is at a low ebb.
For, if I imp my wing on thine,Affliction shall advance the flight in me (19-20)
The speaker has been so weakened and thinned by sin and sickness that he can no longer recover on his own. In order to rise up to God's forgiveness, he must first borrow some of God's strength. But he also claims that his "affliction shall advance" his flight, or in other words, lengthen it. Doesn't that sound like a bad thing? Not so fast. For the speaker, advance = enhance. He welcomes the depth of his affliction because it means that his corresponding trip out of it is even more miraculous and God's greatness and mercy that much stronger. This is the religious paradox the poem ends on: out of great weakness comes great strength.