Although Honor Oak Park is run by LOROL and is served by Overground trains, the rail journey from Honor Oak Park to New Cross Gate, however you do it, is a National Rail journey. As such the National Rail + Underground fares for a Zone 1-3 journey apply. They are £6 cash £4.10 peak / £3.20 Off Peak Oyster.
The comparative fares for an Underground or Overground only journey are £4 cash, £2.90 / £2.50 Oyster.
If you travel from HOP to London Bridge, Charing Cross, Waterloo East or Victoria then it is a National Rail only journey and the fares are £3.40 cash, £2.80 / £2.00 cash.
We are still a long way from having a truly unified zonal fare system in London - a fact that bus users probably appreciate!