Learn French: On soupçonne qu'il y a du favoritisme ici.

In French, soupçonner que (to suspect that) actually expresses a strong belief that something is true, rather than doubt. Because it states a probable reality, it takes the indicative in affirmative sentences. It is the opposite of douter que (to doubt that), which triggers the subjunctive.

How would I change this sentence to say 'We doubt that there was favoritism' to practice the past subjunctive?

Unlike suspecting, doubting triggers the subjunctive. To express 'there was' in the past subjunctive, use the present subjunctive of the auxiliary avoir (ait) plus the past participle of avoir (eu): On doute qu'il y ait eu du favoritisme ici.

Why is it du favoritisme instead of just favoritisme? In English we don't say 'some favoritism' here.

French requires articles before almost all nouns, even abstract ones. Because favoritism is an uncountable concept, you must use the partitive article du (masculine singular for 'some') to show you mean an unspecified amount of it.

Could I use nous instead of on for 'we' in this sentence?

Yes, Nous soupçonnons qu'il y a du favoritisme ici is grammatically correct and means exactly the same thing. However, in everyday spoken French, on is heavily preferred over nous as the subject pronoun for 'we'.