In this paper, we will study smoothability of a weak Fano 3-fold with only canonical singularities which is obtained as an image of a crepant primitive birational contraction from a smooth weak Fano 3-fold. Main part is on a contraction of type III.
Tatsuhiro MINAGAWA. "Global smoothing of singular weak Fano 3-folds." J. Math. Soc. Japan 55 (3) 695 - 711, July, 2003. https://doi.org/10.2969/jmsj/1191418998