The answer to this will depend on the order of the reaction. If the excess reagent is part of the order, then yes, the rate will depend on it's concentration. However, if the order is only with respect to the other reactant (the limiting reactant), then the rate will not depend on the concentration of the reactant in excess. For example, in the reaction A+B ==> C. If it is first order in A, the even if B is in excess, the rate will not be affected by B. The order can only be determined experimentally, and cannot be deduced from just looking at the reaction.