Speed Up Cross Apply Without Index Hint
The problem you're likely facing is around SARGability, namely using the LEFT function in your WHERE clause:

LEFT(s2.RecordKey, 2) = a.Prefix

With that in there, you're stuck running the function for every row and then comparing it. You can't index for that, as-is. Putting the transformation into a CTE, view, or derived table wouldn't help, nor would writing a function to perform the manipulation.

One way around that is to create and index a computed column:

ALTER TABLE dbo.SlowCrossApply ADD LeftTwoPrefix AS LEFT(RecordKey, 2);


Which can be indexed. I'm also changing your index definition a bit:

CREATE NONCLUSTERED INDEX TodayRecords_Filtered ON 
dbo.SlowCrossApply(LeftTwoPrefix, FromToday, Score6)
INCLUDE (Score1, Score2, Score3, Score4, Score5)
WHERE FromToday = 1


Another alternative would be to dump the results of your CROSS JOIN into a temp table:

SELECT     LEFT(s.RecordKey, 2) AS Prefix, CAST(ROUND(sm.ScoreGoal * COUNT(*), 0) AS INT) AS Quant
INTO       #yourmom
FROM       dbo.SlowCrossApply AS s
CROSS JOIN dbo.SmallTable AS sm
WHERE      s.FromToday = 1
AND        sm.ScoreMonth = MONTH(GETDATE())
GROUP BY   LEFT(s.RecordKey, 2), sm.ScoreGoal;


You followed up with:


  ... do you agree with the FromToday [column] being the index key in
  i02TodayRecords instead of RecordKey, ignoring the specifics of the
  query I used here?


In this case, it doesn't matter much. Non-unique nonclustered indexes store clustered index key columns in all levels of the nonclustered index. See my post here: Where Clustered Index Keys Dare.

And:


  Furthermore, since RecordKey is "random", there would likely be a
  great deal of fragmentation that would occur if it were the index key
  as opposed to FromToday which will always be 1. Do those points sound
  valid?


I generally don't worry about index fragmentation, so no. I aim to create indexes to help queries. A fragmented index is much more helpful than a non-existent index, and in many circumstances you'll never notice the fragmentation.

Hope this helps!