The cousins would either be full first cousins or half first cousins? First cousins would share 12.5% of their DNA and half first cousins would share 6.25% ON THE AVERAGE. This is often enough of a difference to come to a conclusion. The caveat is that results sometimes fall in a gray area in between. For instance, I share close to the typical amount with a first cousin (12.9%), but my sister shares only 8.35%, just because of the random way that DNA is inherited.
It would be more clear cut if your mother could find a fairly close connection who would be related only to the second husband (perhaps one of his sibling's children or grandchildren).
It would be preferable to take the test through 23andMe, which reports percentages. If you test through AncestryDNA, they are likely to report a range of possible relationships, which probably wouldn't answer your question. You would need to download the raw data and analyze it yourself with a 3rd party tool.