So I looked at this. I'll tell you what I think and you are of course welcome to take it or leave it. So, the link is to one of these literature reviews that surveys a range of papers. It does not present any new primary data. Their discussion about DNA repair cites one paper that can be found here:
Large-scale COVID-19 vaccinations are currently underway in many countries in response to the COVID-19 pandemic. Here, we report, besides generation of neutralizing antibodies, consistent alterations in hemoglobin A1c, serum sodium and potassium levels, coagulation profiles, and renal functions...
www.nature.com
This paper compares a number of cell parameters, particularly different signaling pathway associated with inflammation, between vaccination and infection. The phrases DNA damage or DNA repair do not appear in the paper, nor can I see that the authors are considering changes to genome integrity even in an oblique way. Rather they do monitor changes to a signaling pathway, NF-kB, that in some contexts, in some cells can, among things, can impact DNA repair mechanisms. But the primary paper does not show that or discuss it.