Phase correction using second order all pass filters/LMS with Wiener-Hopf equations
I actually don't see how the following diagram can be possibly used for phase correction only as it will have an unavoidable amplitude correction as well. An all pass filter by definition will pass all magnitude without modification and only change the phase. The Time Delay block will similarly pass all magnitude without modification with a linear phase versus frequency. The resulting amplitude at the output will be the vector sum of the linear phase component with the all pass component. Since both will have the same magnitude then the sum will be dictated by the phase (for example if there are any frequencies where the phase for each are related by $pi$ radians there will be complete cancellation. The only way the resulting diagram can maintain a constant magnitude is if the all pass filter similarly has a linear phase (which is a trivial all-pass and further would not modify phase to perform phase correction).
To do phase correction you would use the all pass filter directly (this is its purpose) which as explained above is a filter that modifies only the phase response.