Not too much, but too little. A monopsonist that buys at a lower price than where the market naturally clears will remove more producer surplus than the extra consumer surplus it extracts.
In concrete terms, this means something like "the price of a doctors visit is low enough that a nearby clinic closes down. Even though you were happy to pay the higher price for the convenience, and they were happy to provide service at that price, we no longer get to have that experience."
In concrete terms, this means something like "the price of a doctors visit is low enough that a nearby clinic closes down. Even though you were happy to pay the higher price for the convenience, and they were happy to provide service at that price, we no longer get to have that experience."