For an unbalanced fixed effect ANOVA designs a Type II or Type III sum of squares approach is used. The picture's more complicated with unbalanced mixed model ANOVA, I don't know if there's a remedy - this previous post might help:
Unbalanced mixed effect ANOVA for repeated measures
Might be best to run a generalized linear model (GLM) regression with mixed effects rather than an ANOVA.
A mixed model ANOVA isn't necessarily the same as multilevel modeling. The mixed model is a crossed effect model, where each factor contains the different levels of the other factors. In multilevel (or nested or hierarchical) modeling, certain levels of a factor are embedded within one factor but not others (like different subjects within each clinic).