If we ignore air resistance, then weight and time are the only variables needed to determine the power needed for a given acceleration. Force is related to mass through "F=ma", and power is effectively force over time.

Though, neglecting air resistance is a huge deal. A cyclist traveling over 20mph is spending something like 90% of their energy at that point overcoming air resistance. For objects travelling faster, it's a square low of how much more power is needed. Air resistance is a very huge effect. With that said, if travel is dominated by accelerating and decelerating (eg: urban or mountainous environments), then yeah - weight is a huge big deal.