Does pushing the nose tip after septoplasty cause damage to the result?

The septum is repositioned after a septoplasty and often support splints are put in place to keep the septum in its straight position. Trauma or manipulation to the nose directly after the surgery can undo some of the positive changes.  It is important to leave the nose alone after surgery.  Ideally, it is recommended to prevent trauma or manipulation to the nose for up to 6 weeks afterward.

Based your description, you are probably fine.  It was 4 weeks after the surgery, so it should be safe to gently manipulate your nose.  However, I'd recommend that you see your surgeon who can examine the inside of the nose and make sure everything is fine.