There should not be any problems with spending the summer in Connecticut as long as your ex-husband gets visitation
with your daughter. Also, he should be allowed to take her on vacation or have a summer break with her if the order provides for it. In other words, as long as he reasonable access to your daughter during this time period there should be no problems.
The short answer is yes, financial hardship is one of the reasons a court will consider a petition for relocation.
The long answer:
To obtain court approval to remove a child from the state of New Jersey, a person must be able to show (1) a good faith reason for the move; (2) that the move will not adversely affect the non-custodial parent's visitation, and (3) that it is in the best interest of the children to remain with the custodial parent and move out of the state.
Additionally, if moving to a new state would affect the existing custody arrangement, the party seeking to remove the child must also show it is in the child's best interest to move.
Keep in mind that the custodial parent must insure that the parent who still lives in New Jersey has adequate visitation rights. Quite often, the courts will require that the moving parent be required to pay for any transportation costs for the child to go back to New Jersey to visit with the other parent.