Does this setup create time distortion?
If the threshold voltage for the comparator is also derived from that same power supply by means of a resistive voltage divider, then the time will be independent of the supply voltage.  

But if by voltage drop you mean variations during the X minutes period, then yes, the timing will be influenced.



This graph shows the charging without disturbances in the power supply. The green curve is the output of the comparator.



The situation with a dip in the power supply. You can see that the time the comparator output toggles shifts to the right.