A large water change should not create a mini cycle.
The whole point of cycling a tank is to develop a large population of bacteria to convert ammonia to nitrites and nitrite to nitrates. The bacteria are largely attached to the surfaces of rock and sand. A water change even a large one should have little or no effect on the bacteria population.
As for reducing nitrates, I do not know of any chemicals commonly used in aquariums that will absorb nitrates. You can use algae like chaeto in a refugium to reduce algae but that is usually done as an ongoing procedure.
Another method of reducing nitrate is carbon dosing. You add either alcohol, vinegar or use bio pellets in a reactor. This process stimulates the bacteria to multiply and as they multiply to consume nitrates. You use a skimmer to remove the bacteria and in the process to reduce nitrates. The process requires a skimmer that is "oversized" for the system. And like using chaeto, the process is generally used to control nitrate levels in an aquarium and not as a one time reduction.
For your situation, water changes are probably the easiest and cheapest way to go. You can either do a couple of large water changes or a series of smaller ones.