Analog divider using OpAmp

Yes, the divider definitely will affect the gain calculation, because its equivalent source resistance (Thévenin resistance) of 6.23 kΩ is in series with one of your 10 kΩ gain-setting resistors. The overall gain will be -0.2325, rather than the -0.375 that you're looking for.

If you want an overall gain of -3/8, it would be simpler to forget about the voltage divider and just set up your opamp with 20 kΩ input and 7.5 kΩ feedback resistors.

Also, be sure to use an opamp whose common-mode input range includes ground.

You are right about the gain, however there is a small issue with your circuit.

Due to the low values of R4/R5/C2, you will have an undesirable roll off at the low end.

The effective input impedance is set by R4, so the roll off frequency (-3dB) will be:

\$ 1\over{2 \pi R C} \$ = \$ 1 \over{2\pi \ 1k\Omega \ 100nF} \$ = \$ 1591Hz \$ which is way too high for a decent audio amp.

If you change R4/R5 to 10k and 47k, and increase C2 to 1uF or above, then you will have a much lower range. Also don't forget to check the ADC loading at the output also, the same thing can happen there also (see the PIC datasheet for the ADC specs and it's maximum input impedance, as @Andy aka mentions 10k is almost certainly too high, in fact I'm not quite sure what your intentions are with C1 and R6)

Another thing is that if you want to keep the 2.5V offset, then you need to leave C1 out as it will just remove it

One last things is that if you are using a standard electret mic, then 100 Ohms is probably a bit low for R3 (1k - 10k is probably better, but again remember about input/output impedances)