Translation:I need to cook the soup and wash the dishes.
Given готовить is more prepare type cooking and сварить is more boiling type cooking I went for "I need to bring the soup to the boil and wash the dishes" (I tried I need to boil the soup earlier which wasn't accepted either, which I understand as it isn't good English) but I suppose what I don't understand is whether or not сварить is used more for an already prepared soup that you are going to boil to reheat or for the whole process of preparing something like that requires boiling....
In reference to soup (or any other food, whose cooking consist largely of extensive boiling/simmering), варить=готовить (imperfective) and сварить=приготовить (perfective).
Meantime, "to boil"="кипятить" (this word does not imply actual cooking, just boiling) and "to bring something to the boil"="вскипятить/скипятить".
In fact, wash refer as "to cleanse BY the action of liquid" whereas cleaning doesn't really need liquid and tends more to a result where it is "free of ± contamination". I guess "to clean dishes" is misused and it should be "to wash dishes" in all cases. An expression I guess.
I guess its the same thing as using "cleanse" for dishes. I guess it could technically work as "to make something completely clean", but it sounds weird.
English "to clean" focuses on the result, not on the method, and it corresponds to Russian "чистить/очистить" (imperfective/perfective".
Meantime Russian "мыть" describes a process that requires some liquid (usually water) and hence directly corresponds to English "to wash".
You can clean things with a dry brush or a damp wipe; you would need a lot more water to wash them.
And in any case, "to clean dishes" is not a standard English substitute for washing dishes. In fact, the instructions that came with my dishwasher explicitly stated that I don't need to clean dishes before loading them into the dishwasher. Clearly, what it referred to was removing stuck pieces of food and not washing the dishes.