I agree, that's the problem with lack of context. If it's a parent telling a child they have to eat their soup first before they get the main course, then they would use 'your' or 'the'. However, if it's someone talking about the general conventions for order of courses in a typical Czech lunch, then it would be soup in general.
If you don't eat your meat, you can't have any pudding! How can you have any pudding if you don't eat your meat?!
When translated to Czech, the your will be skipped. Jestli nesníš maso, nemůžeš mít (žádný) dezert! Jak bys mohl mít dezert když nejíš maso?
Lovely, this comes directly from your Finnish use of the partitive :D
However, as VladaFu pointed out, it couldn't work with "sníst", because the "s-" prefix here adds a sense of completion - you must eat the whole object of the verb, although the object can be a part: "sníst trochu polévky" (trocha in accusative, polévka in partitive-like genitive) or "sníst misku polévky", etc. On the other hand, "pojíst" always binds with (partitive-like) genitive, because "po-" is a prefix that adds a "partially" or "gradually" meaning, but "pojíst něčeho" is a little old-fashioned.
So the prefix determines which case goes with the verb. When unprefixed (jíst), you can theoretically choose, but genitive is really archaic here - its partitive function has pretty much disappeared from Czech except some fixed expressions. For example, the only way to say "I don't have time" is "Nemám čas (acc.)", while "Nemám času" was the standard a hundred years ago. But, we still say, as an idiom, "Nemám času nazbyt" - "I don't have time to lose/spare".