There is nothing wrong mathematically (nor ethically) with this objective function. However, this objective is weird in a couple of ways.
First, there is no weighting on these which implies you prefer to minimize these terms in accordance with their orders of magnitude. As has been pointed out, the correlation term is likely much larger so your optimization would be tilted toward minimizing correlation.
Second, from a financial perspective, what you are exposed to (in terms of P&L) is covariance, not correlation. If you are trying to minimize correlation in some downside scenario, you should model that (and use stochastic optimization instead of this deterministic setup).
Are dual-term objectives like you have used? Sure; mean-variance or mean-ES optimizations have similar multi-term objectives. Are dual objectives like "minimize $f(X,A)$ subject to maximizing $g(X|A)~\forall A\in\Omega$" possible? Sure; those are multi-criteria optimizations where you have staging or conditioning.
What you have is not exactly a multi-criteria optimization. If you rewrote this as "minimize portfolio correlation for a portfolio minimizing portfolio variance," that would be a degenerate solution -- since there is only one minimum-variance portfolio so the correlation objective would irrelevant.