"Interrupt at" would be used to identify the point at which the interrupt occurred.
"There was an interupt at the conclusion of the speech."
"Interrupt with" would be used to identify what caused the interuption.
"He interrupted the speech with a question."
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She interrupted me with jeering.
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Interruption in the work of like-Ie distraction from work. A break, for example, when a person speaks.