Corporate jet Captain reported being confused by the reconfiguration of Taxiway P and missed the left turn to continue onto the taxiway; and went into the reconfigured closed area as a result.

Narrative

Landed 4R exited Taxiway Q. There was Aircraft Y just to our left ready to taxi. We were told to contact Ground. I made a right turn on Y get off the runway. FO had accidentally put the wrong frequency in for Ground and he had to switch to the current Ground frequency. This took about 15 to 30 seconds. We contacted Ground and was told to taxi Y; P; hold short 31R. Then MDW Ground Control stated there is a Taxiway Papa reconfiguration. MDW had closed Taxiway H and Y west of P. As we taxied on Y approaching Papa I focused on the reconfiguration of Papa and saw Papa location signs in the area that was closed giving directions to continue on Papa. With the reconfiguration; Papa taxiway turns early to the left into the ramp area. I was focused on the P taxiway identification signs in the closed area and taxied straight ahead from Y. The taxiway line disappeared under my nose and I missed the left turn to continue on Papa. I stopped the aircraft and saw temporary lights on the ground on the front left side of the aircraft. I realized my nose was over the reconfigured closed area and I had missed my turn to continue on Papa. The closed area looked just like the normal open taxiways. Without assurance that I could make a left turn and continue on Papa; we called Midway Ground and asked if Operations could look to make sure we could turn. MDW Airport Operations marshalled us clear of the area and we continued to our FBO. We did not contact any objects on the ground.Things that could have possibly helped would have been to cover the location signs in the closed area so as not to confuse directions? The temporary lights in the closed area were not on. If they were on I might have seen the lights earlier. Since the lights are temporary; they are small and very low to the ground. Midway Ground let us know of the reconfigured area on taxi in and [as] we fly into MDW semi regularly; the change to our normal taxiway was different and had confused us.Contributing factors. Busy taxi instructions. Unfamiliar with P reconfiguration. Border temporary lighting in closed area not on; although it was daytime. Closed area appears open and looks like normal taxiway. Complacency with operating out of MDW previously without P reconfiguration. Being rushed after missed frequency switch to Ground.Corrective actions for us. Review taxiway instructions. If confused; stop; ask for clarification. Since MDW Ground alerted to a reconfiguration on P we should have been more vigilant.