Plot hole: Ian Avery-Cooper loses his lottery ticket which is immediately picked up and appropriated by Leonard Corbyn. However, Avery-Cooper immediately reports the loss to the shopkeeper who sold him the ticket. All lottery sales staff are trained in what to do under these circumstances, because it happens a lot - they cancel the lost ticket and issue a new one. Since Avery-Cooper used the same numbers every week this would not pose a problem, but even if he didn't the ticket would be recorded on the seller's computers and could be precisely reconstructed. This would be even easier since Avery-Cooper reports the loss within a minute of it happening and we see that he has the receipt for the sale in his hand. Corbyn's stolen ticket would be worthless and Avery-Cooper would be able to claim his rightful winnings. If nothing else Ian could have simply bought another ticket with the same numbers.
Plot hole: The entire plot hinges on Jonathan's solution of the murder of the first wife of Stephen Belkin. That work had the effect of exonerating Belkin. But, Jonathan's solution wouldn't exonerate Belkin in the slightest. It only explained how the murder was accomplished. It did not eliminate Belkin as a suspect. Even after Creek's solution, Belkin still had means, motive and opportunity. As the only surviving person in the locked room where the murder happened, Belkin would have remained the most obvious suspect. In fact, Creek's solution tended to show that only Belkin could have done it, precisely because the room was indeed locked - Creek never explained how a third person could have got in and set up the killing. but in the transmitted episode, it only occurs to Creek much later that, duh, of course Belkin did it.