Is there sub-conscious error correction in interpreting heard language? My personal experience suggests there is a background process taking place when interpreting a partially heard utterance. The sound is taken in, the conscious part of my mind begins the process of asking for a repetition because of imperfect reception and then a possible interpretation is delivered to my conscious mind for acceptance or rejection. Usually the logic can be seen, how a few words heard imperfectly can be combined to make sense, a partially heard name combined with a pronoun and a mental list of participants narrows the possibilities to one or two, but I don't have a sense of doing this analysis consciously. A second, almost a paraphrase, interpretation is offered up  seemingly from the sub-conscious, "she's asking about Tony's birthday."

If you try to search for linguistics or language and error correction you will find out that there is a lot of concern about teacher behavior during language acquisition. That is not what I want to know about.

I would like to know if there is a subfield in cog-sci that deals with error correction specifically with regard to heard language. Somewhere I can begin a search. Sometimes all you need is the right search terms.
A simple explanation for the phenomenon is top-down feedback. As the bottom-up acoustic/phonological input is coming in, there is top-down feedback based on your knowledge of the language, the situation, and all of the other contextual information, which is helping to constrain or inform your interpretation of the bottom-up signal. A classic example that is close to the example in your question is the phoneme restoration effect (originally demonstrated by Warren in 1970, and studied in depth by Samuel for decades): if a speech sound is replaced with an extraneous sound like a cough, listeners still report hearing the missing speech sound. Importantly, they report hearing exactly the sound that would belong in that context. So a manipulated word like "*eel"(where the * corresponds to the cough), sounds like "wheel" if it is the context "the *eel was on the axle", but the same word sounds like "peel" in the context of "the *eel was on the orange".

This kind of integration of multiple sources of information is typically called "interactive processing". For a review in the domain of speech perception see McClelland, Mirman, &amp; Holt (2006).

References:

McClelland, J. L., Mirman, D., &amp; Holt, L. L. (2006). Are there interactive processes in speech perception? Trends in cognitive sciences, 10(8), 363–369. doi:10.1016/j.tics.2006.06.007

Samuel, A. G. (1981). Phonemic restoration: Insights from a new methodology. Journal of Experimental Psychology: General, 110(4), 474–494.

Samuel, A. G. (1996). Does lexical information influence the perceptual restoration of phonemes? Journal of Experimental Psychology: General, 125(1), 28–51.

Warren, R. M. (1970). Perceptual restoration of missing speech sounds. Science, 167(3917), 392–393.