So, this question is a bit tricky. I think it would depend on the modification. However, we know that in order to reproduce, cells must undergo meiosis to create gametes. The gametes contain the DNA that will be passed onto the next generation. Thus, a modification in one cell during one organisms's life may potentially affect its genetic information if that information becomes incorporated into its genome.
This may potentially be the explanation for the existence of hereditary cancers as a healthy individual may have an onset of a mutation to one of their cells. The cell may fail to perform apoptosis thus resulting in uncontrolled division of the cell, ultimately forming a cancer. This mutated DNA may become incorporated into the genomic DNA, which will then be passed down, even if it's for an asexually reproducing organism. Being asexual may actually increase the chances for the individual to pass down the modified cell DNA, since it will only cross-over with it's own genetic material.