Two companies agree to refund up to $2 million in wake of scrutiny.
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Five-year plan addressed Congress' requirements and tied funding to results, auditors found.
Senate likely to follow suit, but White House raises objections.
But Senate has already backed separate $1.5 billion measure.
The $13 billion put aside will be far short of the total price tag, forcing the services to raid their budget plans to pay for additional costs.
Measure would prevent OMB from implementing current rules on potential privatization of federal work.