Learn Finnish: Satuin lyömään hudin.

This is a predicate chain. The verb sattua (meaning 'to happen to do something') requires the attached verb to be in the illative form of the third infinitive (the -mAAn form). So, sattua + lyömään means 'to happen to hit'.

How does lyömään hudin translate to 'to miss'? Doesn't lyödä mean 'to hit'?

Yes, lyödä means 'to hit,' but the noun huti means 'a miss' (often used in sports or games). The Finnish idiom for missing a shot or a swing is literally 'to hit a miss' (lyödä huti).

Why is the object hudin in the genitive-accusative case (with an -n) instead of the nominative?

This showcases object case inheritance. The object hudin technically belongs to the infinitive lyömään. However, the case of the object is governed by the main verb of the chain. Because the main verb satuin is a normal, active verb with a personal ending (I happened), the infinitive inherits this active nature. Therefore, its total object takes the standard genitive-accusative -n ending.

I remember learning that sattua means 'to hurt.' Why doesn't it mean that here?

The meaning of sattua changes based on the sentence structure. When used alone or with a location case, it often means 'to hurt' or 'to occur.' But when it acts as an auxiliary verb chained to a third infinitive in the -mAAn form, it always means 'to happen to do something.'