Learn Polish: Niech on przeczyta umowę. - Let him read the contract.

The word niech is used to give commands to a third person (he, she, it, or they). To use it, simply put niech before the subject and the regular third-person verb. Here, niech on przeczyta translates to "let him read" or "have him read".

Can I drop the pronoun on and just say Niech przeczyta umowę?

Yes, you can drop on if it is already clear from context who you are talking about. Including on here adds a bit of emphasis, like "Let HIM read the contract," or clarifies exactly who you mean if it was not obvious.

The lesson says niech is followed by a present tense verb, but isn't przeczyta future tense?

Great catch! Przeczytać is a perfective verb, meaning it focuses on completing the action of reading. Perfective verbs take present-tense endings but actually carry a future meaning. Because it uses the grammatical form of a present tense verb, it fits the rule for niech perfectly.

Why does the word for contract, umowa, change to umowę here?

Umowa is the direct object of the sentence because it is the thing being read. Since it is a feminine noun receiving the action, it must take the accusative case, which changes the final -a to -ę.