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A connected topological space $Z$ is unicoherent provided that if $Z=A\cup B$ where $A$ and $B$ are closed connected subsets of $Z$, then $A\cap B$ is connected. Let $Z$ be a unicoherent space, we say that $z\in Z$ makes a hole in $Z$ if $Z-\left\{z\right\}$ is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.