Given the voltage (V) is drop (or potential difference) in charge
This may be the source of your confusion. Current, as your first learn of it, is the movement of charge. But voltage drop is not a "drop in charge" or a "difference in charge". What is briefly called potential difference, is the difference in electrical potential between two points. Electric potential can be said to exist everywhere, including places where there is no electric charge. It is a field. Likewise, potential differences can exist between two points where neither the points themselves, nor the points in between contain any charge.
At this moment my head goes boom, because if we lower potential difference between two charges significantly, the rate of flow of the charge will drastically increase. I cannot fully understand this
Potential difference and current, although completely distinct on a conceptual level, are nevertheless related on a physical level -- through Ohm's law. In many cases, the current through a component (or wire) is fairly exactly proportional to the voltage drop across the component, and this holds true for a wide range of voltage drops and currents. In that case we say that the component has a particular resistance (within a certain range of voltages and currents). Further, we say that this resistance is the ratio between the potential difference and the current.
$$R = \frac{V}{I}$$
Or, rearranged
$$V = IR$$
However, if you keep in mind that this relationship is really a relationship between two conceptually distinct things, and not a "drop in charge" vs a flow of charges, perhaps your head will not go "boom".
If with the constant voltage and power (in formula) I was about to change width of the wire, should the current rise or drop? A wider wire should give us less resistance and therefore more space for charge to flow
If you keep voltage and power constant, then by Watt's Law, \$P = VI\$, the current must also remain constant. Watt's Law, unlike Ohm's Law is valid regardless of the type of component, or the magnitude of the voltage or current.
[Later, you will learn that some components can store energy, and then release it later. You will also learn that there is a kind of current called displacement current, that does not require a movement of charges. However, these are refinements of the concepts, not alterations in the exactness of the relationship given by Watt's Law. That the ratio between current and voltage is fairly constant is an empirical law valid for a range of materials and magnitudes of voltage and current. That the power transformed in an electric circuit is the product of the voltage and current we take to be more axiomatic. Knowledgable people may wonder whether in a particular component the ratio between voltage and current remains constant if the voltage changes. However, knowledgeable people do not wonder whether \$P = VI\$, unless they are questioning some model or theory of physics.]
Returning from the aside, if your voltage is constant, and the only resistance in your circuit happened to be a wire, if the wire were made twice as wide, the resistance would be half as much, so the current would be twice as much. Thus, the power would be twice as much. The power, in this case being the unchanged voltage times the doubled current.
