One of application of Calabi-ansatz is choosing inital metric to run the Kahler Ricci flow, So if you want to using minimal model program and apply Kahler Ricci flow to find canonical metric study of such Calabi-ansatz would be very important
See the paper of
Jian Song, Yuan Yuan , Metric Flips with Calabi Ansatz, Geometric and Functional Analysis February 2012, Volume 22, Issue 1, pp 240–265
See proposition of 3.2 for study of Sasaki-Ricci flow and finding suitable inital metric via Calabi-Ansatz