For me, the weaknesses are not so much with correspondence theory, but with our understandings of reality and our place in it, and the distinction between our experiential reality and whatever it is that is out there that our experience models.
If that isn’t immediately obvious, then consider this.
Science has now demonstrated beyond any shadow of reasonable doubt that we, and the reality within which we exist, are far more complex than our brains are capable of modeling; so evolutionary processes seem to have selected systems that gave useful approximations within the time and energy budgets available.
Thus what we experience as reality isn’t; cannot possibly be. What we experience is a subconsciously generated model of reality that is, in essence, a series of historically selected heuristic approximations that worked for our particular sets of ancestors.
Thus there is a degree of truth in correspondence theory, and it is highly unlikely to ever be 1:1 correspondence in anything other than the most simplistically trivial of situations. Reality seems to be vastly more complex most of the time, and seems to contain many levels of aspects that are not simply probabilistic uncertainties, but contain fundamental unknowables, that may only ever be approximated to some degree of accuracy.
The usual alternative theory of truth is coherence theory, where all things follow from propositions, but successive thinkers from Russell to Goedel have provided fundamental blocks to that approach.
To me, both approaches seem to be fundamentally flawed, as each makes overly simplistic assumptions about the nature of human experience, and this thing we call truth.
Correspondence theory rests on the assumption that we can actually directly experience reality, which modern neuroscience seems to have conclusively disproved.
Coherence theory rests on the assumption that reality always obeys some set of rules, and quantum mechanics seems to indicate that is a very unlikely situation.
It seems that both the fundamental substructure of reality, and the structure of our perceptual and conceptual systems are vastly more complex and fundamentally uncertain than either of the classical theories allow for (both are based upon overly simplistic sets of assumptions, and both can be useful heuristics in some contexts).