Linear Difference Equation and Method of Least Squares
The notation $ \hat{y} \left( k \mid k - 1 \right) $ usually means this is an estimated value of $ y \left( k \right) $ given all the available data up to time index $ k - 1 $.
So generally speaking, this is a prediction of one step in time of the data.

The case above also suggests linear estimation.
Namely, $ \hat{y} \left( k \right) $ is built using linear combination of all the given data.