There are two understandings of monopsony power - each quite different. The first is a raw market power model, comparable to monopoly. This is the one most people are familiar with (if they're familiar with monopsony at all). The second is based on fixed costs associated with labor. In the case of the low wage labor force these are typically hiring and firing costs, although for higher wage labor forces training costs can be substantial. These fixed costs are taken into account in the long run but they are considered sunk in the short run, driving a wedge between marginal revenue product and the wage rate.
This is important in considering what to expect from minimum wages because it's not clear that the two cases will have the same effect (and of course in reality any given employer/employee relationship is going to have a mix of sheer market power and fixed costs in play).
The market power case is fairly simple - a minimum wage is good for wages and good for employment. We're just eating into the monopsonist's surplus, but she has no reason to pull out of the market unless the minimum wage is set too high (which of course is a very real possibility, particularly when the federal government is setting it for a broad array of occupations and jurisdictions).
The turnover cost case is trickier.
At a first approximation, we should not be so sanguine about the minimum wage if most monopsony power comes from turnover costs. In the short run, yes, firms will increase employment in the form of reducing separation rates (assuming most of the costs are on the hiring end) and raises will rise and everything will be great. But in the long run there's a much greater incentive for reducing employment - maybe.
There is also an incentive to reduce turnover or invest in human capital to make up the difference. That's the thing about fixed costs - you don't have to cut jobs. Since they're associated with hiring and firing you can also amortize them over longer periods of time. So in this monopsony model we have the potential for something bad - reducing employment in the long-run as a result of the minimum wage, or something really good - increasing employment and job tenure for low wage workers.
So monopsony makes the minimum wage waters murkier than a lot of people appreciate. And the monopsony argument is more complex than a lot of people appreciate.
This makes testing on multiple outcomes - long term employment effects, turnover rates, on the job training rates, etc. very important.