Learn Portuguese (Portugal): O artesão quase que deixou cair a haste.

This is known as a partícula de realce (emphasis particle) or expletive que. It doesn't change the translation at all, but native speakers frequently add it after quase to emphasize how close the action came to happening. It makes the sentence sound slightly more dynamic or dramatic.

Can I just remove the que and say O artesão quase deixou cair a haste?

Absolutely. Because the que is just an emphatic particle here, removing it leaves a perfectly correct and natural sentence. You will see both versions in Portuguese, though keeping the que is very common in everyday spoken language.

Why use deixou cair instead of a specific verb for 'to drop'?

Portuguese doesn't have a single, common direct equivalent for the accidental meaning of 'to drop'. Instead, it uses the verb deixar (to let) followed by cair (to fall). So literally, the artisan 'let the rod fall'.

What exactly is a haste?

A haste is a versatile noun used for a long, thin support structure. Depending on the context, it can translate to a rod, a metal shaft, the stem of a flower, a flagpole (haste da bandeira), or even the arms of a pair of eyeglasses.