Translation:We have an important job to do.
For what it's worth (a lingot?), I agree with you. Realizar carries a sense of termination.
Is the "a" required only because of the adjective (importante)? Would "temos um trabalho realizar" be correct, or would the "a" still be required?
It would be still required because it shows the purpose. So, both para and a work here.
How about "We have to accomplish an important work"?
Should it be accepted as "accomplish" is also shown in the hint or is only "to do" valid?
Google translate suggests for reverse EN-PT: "Temos que realizar um trabalho importante".
There is a distinction between "we have + noun" and "we have to + verb":
We have an important job to do. -> Nós temos um trabalho importante a realizar.
We have to accomplish an important job. -> Nós temos que realizar um trabalho importante.
Ahah lol. The previous sentence "I tenho um trabalho importante a realizar" I translated as "I have TO DO an important work", Duo said it's wrong and suggested "CARRY OUT", now it was quite opposite)))