Error correction code without error detection
You might be interested in the binary erasure channel, in which each symbol is erased with probability $p$. The capacity of this channel is $1-p$, and there are practical erasure codes that achieve it.
In the second scenario you describe, some symbols are erased, and some are received with error. This is known as the binary symmetric error channel, and there is some work on it.
Finally, if each symbol is associated with a confidence, then the corresponding notion of decoding is soft decoding.