Not accurate readings when connecting OpAmp AD822 to ADC MCP3008
The voltage over the 10k will always be practically 0V, because no current is flowing in the ADC input. You have to measure the voltage over the capacitor (meaning from ADC input to ground) to see the same voltage the ADC is seeing.
The fluctuations you see in the ADC signal indicate that the OpAmp is not in stable operating conditions. As Andy mentions in the comment: You need to decouple the supply voltage with something like 100nF between V+ and V- as close as possible to the supply pins of the OpAmp. This gives a low impedance path for the supply current.

Another common cause of instability with transimpedance amplifiers is a lagging phase due to the capcaitance of the photodiode. If the phase lag approaches 180° the negative feedback becomes positiv feedback and the amplifier may start oscillating.
To compensate for that a feedback capacitance \$C_f\$ is placed in parallel to \$R_f\$. In your case that is the 1300pF. How did you get that value? You need to know the capacitance of the diode as well as the GBW of the OpAmp to calculate a matching \$C_f\$. You can read the details in this TI paper Compensate Transimpedance Amplifiers Intuitively, but the short form is:
\$ C_f = \frac{1}{4 \cdot \pi \cdot R_f \cdot GBW}[1 + \sqrt{1+8 \cdot \pi \cdot R_f \cdot C_D \cdot GBW}] \$
So if your feedback capacitance varys to much from the ideal value that might be the cause of the instabilities you see.