Yes, "biurko" is singular, "biurka" is plural, but "biurka" is also the genitive form of singular (see https://en.wiktionary.org/wiki/biurko). The verb "potrzebować" is used with the genitive, hence "To jest biurko. Potrzebujemy biurka." = "This is a desk. We need a desk". In plural: "To są biurka. Potrzebujemy biurek" = "These are desks. We need desks".
Is there any pattern or rule of thumb for which verbs use the genitive? This is pretty confusing learning new nouns and new verbs at the same time
Yes I wish they'd include this with the notes or something! I can look up how to use the different tenses somewhere else but there's so many different situations where you use one tense or another it's hard to find a good explanation on when to use them.
Well, until then... all hail mowicpopolsku.com haha
It seems that genitive is for negative verbs. So anything that's Nie (verb) is genetive. And also anything that implies you are lacking something like potzrebujemy "We need" implies that you don't have the desk so it's negative. If it was "Szukamy" "We are looking for" it would be the same.
Such logic may be rather problematic. First of all, "nie" doesn't automatically mean Genitive. It may be your first association, because you see Genitive in a lot of negations, but it's not for everything.
In fact it is like that: if a positive sentence needed Accusative, it takes Genitive instead when negated. "Lubię koty" vs "Nie lubię kotów". But this is the only case that changes when negated. All others stay the same.
Moreover, it's not like a similar 'topic' makes the verbs take the same case. Sometimes there isn't really much logic to it. Similar verbs meaning (almost) the same can take different cases sometimes. The example that comes to my mind is "używać" (to use) that takes Genitive, but "posługiwać się" (which probably would also be translated as 'to use') takes Instrumental.
So I'm afraid that why it may be useful to have this association, you can't treat it as a rule.