Found the solution. The following MDX gives the values that I want.
SELECT {MEASURES.[Avg Test Score],
%LIST(NONEMPTYCROSSJOIN([BirthD].[H1].[Decade].Members,{[Measures].[Avg Test Score]}))} ON 0,
NON EMPTY homed.city.MEMBERS ON 1
FROM patients
I think the key was to enclose the measure in the NONEMPTYCROSSJOIN function in curly braces. I had not done this in a previous attempt at getting this to work.
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