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Your first circuit topology makes sense, although some of the part values are questionable.

<h2>How the circuit works</h2>

R37 provides the bias supply for the condenser microphone with a predictable impedance.  C33 tries to reduce noise on the 3.3 V supply.  

The output of the microphone is AC-coupled to the amplifier thru C28.  Since audio has no content below 20 Hz, eliminating DC (0 Hz) is a valid thing to do.

R28 and R34 divide the 3.3 V supply to make the DC bias voltage for the signals around the amplifier.  C29 filters out noise from the 3.3 V supply that could otherwise make it onto the bias voltage, and provides a lower impedance to the AC audio signals than the resistor divider would by itself.

R36 couples the bias voltage onto the audio input of the amplifier.  C28 blocks DC and the opamp input is effectively infinite impedance from this circuit's point of view, so R36 can set the DC bias point while loading the AC signal relatively little.

C21 and C23 are power supply bypass caps for the opamp.  They try to keep the 3.3 V powering the opamp reasonably clean.

R27 and R26 are the negative feedback divider.  Since 1 part in 101 is fed back onto the opamp negative input, the overall amplifier gain is 101, assuming the usual conditions around negative feedback gain setting are met.

C24 adds impedance to R26 at low frequencies.  This reduces the overall amplifier gain at low frequencies, to the point that it becomes only 1 at DC.  This keeps the inevitable opamp input offset voltage error from getting amplified, which could push the output DC bias point so far to one end of the output range that signals with reasonable amplitude could clip.

C32 AC-couples the amplifier output to the overall circuit output.  This makes the amplifier internal DC bias point irrelevant to the outside, and allows whatever follows C32 to the right to set its own DC bias voltage.

<h2>Part values</h2>

As I said before, the circuit topology is reasonable, in fact common.  However, some of the part values don't make sense assuming this for normal audio signals with a maximum range of 20 Hz to 20 kHz.

R37 is a function of the microphone.  You look at the datasheet to see what kind of impedance this particular microphone wants to work with.  100 k&Omega; might be good if this really is a condenser microphone, but should likely be much lower if it is an electret.

3.3 V is very little for a true condenser mic.  It is in the normal range for electret mics, which is why I wonder if this circuit isn't intended for an electret despite what the label for MK21 says.  Only the microphone datasheet can say for sure, but one of the supply voltage and the R37 value are probably not right.

C33 may not be doing much, since it's action is dependent on the impedance of the noise signals on the 3.3 V supply.  I would add a deliberate resistance between it and the supply.  If sticking with 4.7 &micro;F, then 3.6 k&Omega; would cause a rolloff frequency of 9.4 Hz.

Taking R37 of 100 k&Omega; at face value, that is the impedance of the audio signal coming directly from the mic.  A value of 100 &micro;F for C28 makes no sense.  Again, the lowest valid content of even "HiFi" audio is 20 Hz.  Since there will be multiple places where we have high pass filters, lets set them to 10 Hz or so.  The 10 Hz rolloff capacitance at 100 k&Omega; is 160 nF.  A 200 nF cap would make sense.  100 &micro;F (500x more) is totally absurd and will cause the circuit to take a very long time to stabilize from power up.  200 nF also allows use of more sensible cap technologies than tantalum.

R28 and R34 make about &frac13; of the supply as the bias point.  Why not &frac12;?  You want the bias point to be about in the middle of the available output voltage swing.  Assuming U26 is a CMOS rail-to-rail output opamp (I didn't look it up since you didn't provide a link), &frac12; of the supply would make the most sense.

The impedance out of the R28/R34 divider is 32 k&Omega;, which is relatively high compared to the 100 k&Omega; impedance of the audio signal.  Maybe there is a need for compromise here due to battery operation, but otherwise I'd want the divider output impedance to be lower.

C29 should have about a 10 Hz rolloff with the divider output.  That would be about 500 nF.  The 10x higher you have makes the circuit startup time longer than necessary.

R36 should be several times the impedance of the audio signal.  Being equal to the signal impedance, you are losing &frac12; the signal amplitude thru R36.  Assuming U26 has high impedance inputs, the only downside to making R36 larger is longer circuit startup time.

R27 and R26 set the gain to 101 assuming a perfect opamp.  That's quite high.  If you want the circuit to work over the HiFi audio range, then you are asking for a gain of 101 at 20 kHz, which is a gain &times; bandwidth of 2 Mhz.  Due to relying on negative feedback, the opamp itself needs to have a gain &times; bandwidth 5-10 times that, or 10 MHz minimum.  I didn't look up the opamp, but that sounds high for something that needs to be low noise.

A gain of 10 or maybe 20 would be more reasonable.  That would allow C24 to be lower, which is desirable to avoid using a tantalum cap.

The polarity of C32 is questionable.  If the output is meant to have 0 V DC bias, then it is definitely backwards.

<h2>The second circuit</h2>

This circuit differs only in that the feedback divider reference point is the bias supply, instead of being independent.  I guess that's a possible simplification that saves one capacitor.  Personally, I'd go with the first circuit, other than the brain-dead values as already pointed out.