Can temporary denial of oxygen to a cell damage the cell? Will the cell continue to live in partially damaged condition if the supply of oxygen and nutrients are restored?
This is really interesting because the answer largely depends on the severity of the damage, the length of the damaging condition, the type of damage incurred, and the type of cell incurring the damage.
For a real life example, irreparable DNA damage as by irradiation or drugs tends to be lethal to a cell. The severity of the damage may affect how long a cell has to live, however. For example, PBMC we irradiated with X gy tend to stay alive for 10-15 days (they actually don't divide though, due to the damage). Why did we pick that number? Too low a dose the cells persist, and too high you obliterate them. For reference, my experiements require my PBMC to be alive for a few days but disappear from the culture over time, so X gy is just an optimized value. Gy (gray) is an SI value for dose of ionizing radiation.
What you call a temporary denial of oxygen is actually referred to as intermittent hypoxia, or in a prolonged case just hypoxia. Human cells regularly survive in hypoxia by altering their metabolic pathways when OxPhos can't be utilized anymore. These changes are assisted by oxygen-sensing molecules that change the transcriptional profile of the cell. HIF1a comes to mind, a hypoxia inducible factor, which is regularly induced by PI3K and MAPK (typical cellular signals). It's typically kept in check by other oxygen sensitive proteins like VHL, but in hypoxia the effects of HIF1a action are entirely beneficial. So when the cell returns to "normoxia" it can typically reprogram into OxPhos (caveats exist, tumors for example).
So to answer your question yes, the cells will survive (in nutrient deprivation, too, but with a different mechanism), but they have to make a lot of changes first.