Learn Polish: Podobno mnóstwo studentów ma sporo problemów z tym tekstem. - Supposedly plenty of students have quite a lot of problems with this text.

When indefinite numerals or quantity approximations like mnóstwo (plenty) or sporo (a lot) act as the subject of a sentence, the verb must take the third-person singular form. Even though we are talking about many students, the grammatical subject is the singular concept of a multitude, so we use ma instead of mają.

What case do the nouns take after approximation words like mnóstwo and sporo?

They require the noun to be in the genitive case. Because we are talking about countable things in a large quantity, we use the genitive plural here: studentów (from the nominative student) and problemów (from problem).

Is there a difference in scale between the approximations mnóstwo and sporo?

Yes, while both words estimate quantity without giving an exact number, they differ in scale. Mnóstwo is a very strong word meaning a huge amount, a multitude, or loads of something. Sporo is a more moderate approximation meaning quite a lot or a good amount.

We see z tym tekstem here. Does the noun problem always take the preposition z and the instrumental case?