Does the magnetic field- due to the current carrying wire- curve in a clockwise or counter-clockwise direction in the area between the poles of the horseshoe magnet? how do you know?

The direction would be dependent on the direction of the current through the circuit. But for this, you can use the right hand rule to determine the direction of the magnetic field. The right hand rule states that: to determine the direction of the magnetic force on a positive moving charge, ƒ, point the thumb of the right hand in the direction of v, the fingers in the direction of B, and a perpendicular to the palm points in the direction of F.