Learn Swedish: Forskaren har redovisat studien som visat skillnaden. - The researcher has presented the study that has shown the difference.

This sentence demonstrates a grammar rule called ha-bortfall (omission of 'ha'). In Swedish, you can often drop the auxiliary verbs har and hade when they appear in a subordinate clause. The clause starting with som is a subordinate clause, so som visat is completely correct and still means "that has shown".

Can I also drop the first har and just say Forskaren redovisat?

No, you cannot. Dropping har or hade is strictly a rule for subordinate clauses. Forskaren har redovisat is the main clause of the sentence, so the auxiliary verb is mandatory here. If you drop it, the sentence becomes grammatically incorrect.

Should I use this omission trick when I'm speaking Swedish?

You can, but you don't have to. Dropping har and hade in subordinate clauses is very common in written texts, news, and formal Swedish. In everyday spoken Swedish, it is actually more common to keep the verb and say som har visat skillnaden.

What is the difference between redovisa and visa?