Barker quit the Shakers to move to the Broadfield Stadium at the start of the season and has guided Crawley to sixth in League One.
But they have just lost back-to-back matches at Brentford, in the league on Saturday and in the Johnstone's Paint Trophy three days later.
Red Devils boss Barker said: "It is disappointing to go out of any cup competition but we now have four league games - three of them at home - before the FA Cup and it's time to crack on with the league.
"We don't like losing two in a row but Brentford are a decent side. It's back to the Broadfield on Saturday and we will be looking to put things right against Bury."