As being minimalist, the following single line will also work to set your param to null if it is a blank (or spaces) without any trims and checks for length (it would also work if your @path param would be of CHAR datatype):

SET @path = NULLIF(@path,'')

Actually, you don't need even that, just use:

UPDATE m

SET

Path = ISNULL(@Path,NULLIF(m.path,'')), -- don't update when no path supplied

PathReadOnly = ISNULL(@PathReadOnly,m.PathReadOnly)

FROM dbo.Mytable as m

WHERE Code = @Code