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Yes you do need to take his statements seriously.
Regarding the benefits fraud both you and your exhusband could be found guilty of the fraud if raised by either of you. He would be guilty or the recertification and you would be an accomplice by allowing it to happen since they are in your name.
Normally a court will give preference to a biological parent over a nonbiological parent for custody matters but the court can deviate from the presumption if it feel the deviation is in the best interest of the children. In this case since there have been a very long history of the children living with your exhusband, the fact that they continue to live with him instead of you after you moved and their age would make it possible for your exhusband to seek custody. If he obtains custody it is possible the court would review the child support issue.