Unexpected opamp output
Post Mortem ( after answer accepted)

The real problem with this question is the user copied a design without understanding what every part does and then realize not every design you find is perfect. ( user needs to read more App notes until understood )

Lessons to Learn

Perfection in Engineering IMHO means meets or exceeds all specs or measureable requirements (cost, performance, quality metrics) but if you dont have any or don't know what is important, Warning! this is not Engineering rather a casual hobby interest.  The best designs simply meet the best specs.  Often when pressured for schedules we must accept this criteria even if we wanted to do more and so we think it's only 90% complete. So learn to make hard decisions on specs.

The Mic has an active JFET open drain so the pullup resistor of 10k determines AC voltage gain and DC operating point and if matched perfectly will be mid-scale on Op Amp output of full output swing or Vbat/2.

This makes the Vbat/2 buffer circuit in the green square completely irrelevant* along with 100k resistor.  The Gain ratio is AC coupled by 4.7uF in series with 100 Ohms and drops -3dB at 340 Hz and ought to be 5 to 10x bigger for better bass response.

Why*?

Since the the 4.7uf feedback cap (or bigger!) blocks DC but needs low impedance due to 100R it can now go to ground and no longer needs the 100K bias R to V+/2. With matched input series resistors of 10K from FET pullup to Vin+ and 10k Feedback to Vin- the input bias currents produces the same voltage drop thus cancelling but since the DC gain is 1+1 we should expect the output to be roughly V+/2 within 10% or less.


  It is also important to verify leaky caps.


Removing the 4.7uF proved that but then when you saw the half wave you doubted it. Consider that during the input zero crossing, the output ought to be at the same as the DC  voltage 4.5 or so. 

final

To determine AC gain. define impedance and minimum used voltage input  and available output swing. Then define load impedance and choose high Z headphones (e.g. 300 Ohm Sennheiser ) for lower power or 60 Ohm Buds but if wanting to use a 9V battery , 8 Ohms will drain the battery fast  Most likely. 40dB gain is too high.  Examine any high quality audio amp specs and consider at least a dozen critical specs in your next design.