Circuit to switch off circuit "B" for 2-3 seconds when circuit "A" is switched off
The following circuit should accomplish what you want:

Normally pin 3 of the 555 timer is off, which keeps the P-channel MOSFET Q1 (high side switch) on, which powers circuit B (output on the left).

When circuit A (input on the left) is powered off, the line going into the trigger input (pin 2) of the 555 is monetarily brought low (C1 and R1 act as a differentiator) and triggers the 555 configured as a monostable.  The output pin 3 goes high for 2.5 seconds (determined by the combination of R1 and C2), then pin 3 goes back to 0, turning the MOSFET back on.

The MOSFET is rated at 5A.  If for some reason you can't drive circuit B from the MOSFET directly, a lower-amperage MOSFET could be used to operate a relay.