Nest your JSON calls:

SELECT MT.Id,
       JT.InitialCost,
       JT.OtherCost,
       DV.value
FROM #MyTable MT
     CROSS APPLY OPENJSON(MT.MyJsonText)
                 WITH (InitialCost decimal(8,2),
                       OtherCost decimal(8,2),
                       DatesOfVisit nvarchar(MAX) AS JSON) JT
     CROSS APPLY OPENJSON(DatesOfVisit) DV

Nest your JSON calls:

SELECT MT.Id,
       JT.InitialCost,
       JT.OtherCost,
       DV.value AS DateOfVisit
FROM #MyTable MT
     CROSS APPLY OPENJSON(MT.MyJsonText)
                 WITH (InitialCost decimal(8,2),
                       OtherCost decimal(8,2),
                       DatesOfVisit nvarchar(MAX) AS JSON) JT
     CROSS APPLY OPENJSON(DatesOfVisit) DV;