If your only purpose is just to test if there is a significant relationship between your dependent variable and a set or a subset of predictors, then the actual performance and explanatory power of the whole model may be neglected, as long as you verify that the model as a whole is significant.
However the more important point here is the following. If you know or reasonably expect that a predictor like the health status is significant and you force it outside the model, then this may result in a bias on the parameters of the retained independent variables if the excluded important parameter has some non-insignificant correlation with the retained parameters. See for example this source for more details on this point. So my best advice is to include the health status even if you are not directly interested in measuring its significance (it will be retained as a control variable, so that the model can assign to it its own coefficient and clean up the other remaining coefficients from the possible biases due to the correlation between those variables and the health effect). Then, if it is not a subject for your research, you are free to choose not to test it and neglect it. But, if, as you say, that parameter is important (the data will verify it!), then the model will assign to it its own coefficient, which is good for the estimation and the standard error of the other parameters.
