A cheap psu may not deliver advertised power.
If a 600w psu does not have at least 2 6 pin pcie adapters, it is suspect.
I would consider replacing the unit with a better quality psu like Seasonic.
A 6870 is older tech and less efficient. But, it still should run in a good 500w psu.
If you overprovision a bit to a 620w Seasonic, you will have the ability to upgrade to any single gpu card as strong as a R9-290X or GTX780ti.