Why does my spinach pasta break when I run it through the cutter?
In terms of the science of the dough, there are three probable culprits to consider:


Insufficient hydration.  That is, your dough either doesn't have enough water (from any source, including for example, any eggs) to form a cohesive dough, or it has not rested long enough for the water to be absorbed into the starches of the flour.

The symptom for this is that the dough is in general crumbly and frangible, and doesn't want to hold together.

If this is the case, the remediation would be simple:  add more water, and let the dough rest about 30 minutes.
Insufficient gluten development.  Gluten is what gives pasta its strength.  It needs both enough water (see issue number 1), and enough kneading to develop.  You may need to knead your dough more.

If this is the case, knead by hand until it is cohesive enough to go through the rollers.  The first rolls not only thin the dough, they continue to knead it, further developing the gluten.

The symptom for this is that it holds together when pressed, but not when worked, compared to the first issue.
Excessive interference.  There may be too many other ingredients (such as the spinach powder) getting in the way of the gluten formation (see point number 2).  They can physically impede the gluten from developing.   You only want to add an accent of non-flour/water ingredients.

This would be the hardest case to recover from, as you essentialy would need a new recipe with a new balance.  You could try making a second batch without the interfering agent, and combining them, but probably not worth the effort.

It is hard to tell if this is the case, versus number 2, unfortunately.  If you are using a quality recipe, this is unlikely.