Consider the following experiment: from a lacquered copper wire we cut off twenty to thirty pieces of about 10 cm. From them we form a bundle of parallel wires and connect the two ends with one more wire each. The other ends of these two wires are connected to a sensitive analog ammeter. We hold the bundle horizontally and move quickly a strong and broad magnet downwards on its left side. The pointer of the instrument will make a deflection to one side. If we now move the magnet quickly downwards on the right side of the bundle, the instrument will make a deflection to the opposite side. The magnetic flux that we have produced in the wire is now in the opposite direction to the one in the first case, which is why the deflection is in the opposite direction. The motion of the magnet produces current even if we only approach it to the bundle from one side without lowering it below the bundle. In this case the current is somewhat weaker. But if we now move the magnet down to the middle of the bundle, the instrument won’t show any current, because the left and the right halve of the magnet act on opposite sides of the bundle, canceling each other out.
We can do the experiment with only a single wire instead of a bundle, as long as we have a very strong magnet and a very sensitive ammeter.
You can imagine that inside this wire there is a propeller or there are many propellers in a row. When you turn a propeller manually from the left side, then it is turning in one direction and it is blowing on one side (plus), but it is suctioning on the other side (minus). When you turn the propeller from the right side, then it is turning in the contrary direction and the air current is in the opposite direction. But you cannot turn the propeller from above. Exactly the same picture we have with the magnet and the wire. (You can read more about this in my book https://newtheories.info  pages 38–40).

Now back to the question.
Let’s say we have a battery whose poles are connected with a wire. Current flows through this wire. If we quickly lower a magnet in the vicinity of the wire from one side, we can for a very short moment increase or decrease the current in it, depending on that, whether we do it on the left or on the right side of the wire. With ordinary magnets and only a single wire this effect will be extremely weak.

Please read also Inducing electric current in a wire by moving magnetic field