In a great many instances you can omit "that" and your sentence is still fine. In the above sentence, though, omitting it makes it sound just a bit clunky and awkward. In sentences that begin with phrases like "it is thought that...," "it is hoped that...," "it is believed that...," etc, the "that" is pretty much always retained. So in answer to your question whether it's possible to omit it, honestly I don't know. You should keep it though, just because it feels better.
Also, "it is hoped that the fish will help the exact source and spread of any pollution to be better understood."
(not a teacher, just a language lover)