grudge one's eyes time enough?

Do you mean gouged?

grunge does not make sense

grudged = allowed under sufference.
I grudgingly allowed the kids to eat ice cream.
Grudged is not a common usage to my knowledge

other useages
to hold a grudge
a persistent feeling of resentment, esp one due to some cause, such as an insult or injury

It means that the author was reluctant to stop looking around, to stop seeing the desert. Grudge my eyes time enough to wink means the author did not want to close his or her eyes even for a split second.

Thought that was "begrudged"

Both are acceptable.