5 A current sensor ACS712 has sensitivity of 185 mV/A, i.e for rise of 1 A it can produce 185 mV at its output. Can we increase it to, say, 1 V/A i.e. for rise in 1 A of current it can produce 1 V at its output?
For the provided information, maybe this configuration at the IC output provides a solution (in the linked image):
The output will be the input amplified (1+R2/R1) times. Regarding the the operational amplifier, as long not too low voltages (i.e lower than 50 mV) or too high frequencies (depending on the desired output voltage) are involved, any model can fit the circuit (TL082 best option).
Yes, you can by amplifying the output from the IC by the factor of 1/0.185 = ~5.4. But keep in mind that you will also be amplifying the offset errors and the output referred noise by same factor, which may or may not be acceptable in your application. For example if the output noise is 21mvpk-pk as in table on page 5 of data sheet, the noise will be increased to 5.4x21mVpk-pk = ~113mVpk-pk at the output of the amplifier (the amplifier will also add it's own noise by the noise gain of the amplifier x amplifier input noise). In this case you may need to further limit the circuit bandwidth, if acceptable, to reduce the noise. But this depends on the needs of your application. See application circuit 3 in data sheet for an example of adding external gain...
You can't 'tune' the ACS712. And adding an amplifier is not trivial considering the VCC/2 output for 0 A. I'd suggest to search for another sensor that better fits your needs. There's a lot of ICs resp. current sensor assemblies available.
I agree with U. Dreher above. This circuit is for measuring anps. There are circuits for measuring milliamps, microamps, nanoamps, or picoamps. What are you trying to measure? A switched charge amplifier into a differentiating stage, has sensitivity to the lowest currents that I have measured.