The phrase "mortal flesh" refers to the human body that is susceptible to illness, aging, and death. It emphasizes the temporary and fragile nature of human existence. Full definition
Before death, while we are still in mortal flesh, we eat the labors of our hands, we swallow with an effort the food so gained; but after death, we shall begin eagerly to drink in the spiritual life and finally, reunited to our bodies, and rejoicing in fullness of delight, we shall be refreshed with immortality. (firstthings.com)
In addition to this, there is another interesting insight: the role of chastity as the healing or «remedying in our own mortal flesh of the wound of concupiscence.» (faith.org.uk)
After Adam sins and death enters the world, weak flesh becomes mortal flesh. (firstthings.com)