Can mitochondria with "depleted" mtDNA reproduce?
Can mitochondria with "depleted" mtDNA reproduce ?


If by depleted, you means reduced number of mtDNA copies, then yes, as long some copies still remain, these can be replicated. 

Mitochondrial genome is replicated asynchroneously from nuclear genome [Bogenhagen, 1977], through a particular process (review [Holt, 2012]), and is present in multiple copies. These copies can be different, due to mutation.

Mitochondria do not "reproduce" in the common sense, as they do not undergo mitosis, but have a fission and fusion mechanism (review [van der Bliek, 2013]). The fission products get their half of the mtDNA copies at random when the two mitochondria separate.

Mitochondria with depleted mtDNA can (and do, according to your image) undergo fission.


  passing on the damage from breaks in the DNA? 


There are DNA repair mechanisms for the mitochondrial genome. In this case, the mechanism of choice to repair double strand break is  microhomology-mediated end joining (MMEJ) [Tadi, 2016], that try to align the ends of the DSB through use of 5-25 bp homologuous sequence. This ususally results in the deletion of some material near the break.

The resulting mutation is conserved, and passed on through replication of this particular mtDNA instance. The mitochondria will contain a mix of mutated and non-mutated mtDNA from now on.


  Or is the mitochondria and cell doomed?


One could say a particular mitochondria is "doomed" if there is no [viable] copy of mtDNA; if it is not able to maintain adequate internal membrane potential, it will try to fuse with other working mitochondria, or be eliminated through autophagy/mitophagy.

The cell itself could be "doomed" if there are not enough mitochondria left to meet energetic demands. (or, through mitochondria-induced aptotosis). Some particular human cell type have hypersensibility to fluroquinolones (tenocyte, chondrocyte, osteocyte), with increased cell death, but other types have not manifested this behaviour. [Barnhill, 2012] The reason for this sensibility/resistance is not elucidated.