You can join your query with the Data
table:
SELECT DISTINCT ON ("TrainingMatrix".payroll, "TrainingName", "Institute") "Data".EmployeeName, "TrainingMatrix".*
FROM "TrainingMatrix"
JOIN "Data" ON "TrainingMatrix".payroll = "Data".payroll
ORDER BY payroll, "TrainingName", "Institute" ,"TrainingDate" DESC NULLS LAST;