Most higher education facilities are cleaned by in-house cleaning staff, an outsourced cleaning contractor, or a mixture of both. Whoever is doing the cleaning, the worst thing that anyone in charge of a cleaning staff can hear are complaints. In fact, that might be the only time you hear from higher-ups, your supervisor, or your client. It's just a part of the job.
When it does happen, though, it’s important to act and see why the client is unhappy. There might be training issues, of course. But, it’s entirely possible that the cleaning staff is doing everything that’s expected of them. That, however, might be your issue.
One thing you might find helpful to ask: Is everyone on the same page?
In other words, do you and the university staff have the same expectations as to what you're meant to be cleaning? The truth is, the cleaning needs of a facility are continually changing. New laws go into effect. Schools are remodeled. New kinds of facilities are erected.
All of this can be managed by making sure that your staff's "scope of work" is updated regularly. If you don't have a scope of work, then make sure you obtain one or make one. A scope of work will go over all the areas in a facility, and detail the cleaning expectations, and how often they should occur. A snippet from a scope of work might look like this:
Some things to consider when working on a scope of work: