How can I move part of a mesh along the right orientation? When I use a mirror modifier for my mesh and try to move something, the normal orientation distorts the mesh.

It turns this


into this


When it should turn it into something more like this, but with greater precision.


How can I fix it, so that I can move the part of the mesh across the right orientation?
IMO, a convenient solution is to use two 3d windows so you can have multiple views simultaniously. The left window is oriented is in your picture, the other is an orthogonal view (left, front, top, bottom, back right). In the latter view (highlighted), use the red and blue arrow to move: