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We show that there exists a positive real number $\delta>0$ such that for anynormal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worsethan canonical singularities, that is, the minimal log discrepancy of $X$ isless than $1$, then the minimal log discrepancy of $X$ is not greater than$1-\delta$. As applications, we show that the set of all non-canonical kltCalabi-Yau $3$-folds are bounded modulo flops, and the global indices of allklt Calabi-Yau $3$-folds are bounded from above.

A gap theorem for minimal log discrepancies of non-canonical singularities in dimension three