It is worth noting that on these dates, and for much of the season, only half the service to Whitby will be steam hauled, the other services are using heritage diesel locomotives.
Also in early season and most Sundays throughout the year some of the Pickering to Grosmont services are being run by the preserved autocar.
That is a wonderful piece of railway history- a petrol/electric train- as far back as the start of the 20th century the North Eastern Railway recognised that steam traction was not the future for passenger traffic on lightly trafficked branches and developed the autocar- a (then) petrol driven multiple unit which generates it's own electricity for propulsion.
Far more economical to run- https://electricautocar.co.uk/
Also on the Tuesday Laurel is interested in, if the bridge repairs are complete, they are running the North Yorkshire Pullman dining train- 1235 from Grosmont to Pickering, returns at 1520-
https://www.nymr.co.uk/Handlers/Download.ashx?IDMF=ea71839b-b972-4522-9e90-f07a6f8de16b
It is, without question, a splurge, but is an option.
As the Pullman stock is based at Grosmont (and currently stranded there due to the bridge failure) I'm not sure what they will do if the problem persists.