New York Gov. Kathy Hochul announced Tuesday as part of her budget address that she plans to introduce legislation to apply a bell-to-bell restriction on smartphone use in schools statewide. Many of the city’s more than 1,500 public schools already restrict cellphone use, but Hochul says a listening tour through New York’s educational system highlighted, to her, the need for additional measures.    The complete ban on usage would mean no sneak peeks of student cellphones in school cafeterias, halls, or bathroom stalls. The Democrat says the ban on school grounds is necessary to make the learning environment distraction-free and to preserve student mental health. Details on potential violation penalties weren’t immediately clear. Hochul proposes investing $13.5 million to help schools implement the new plan, which would ban unsanctioned use of any Internet-enabled device on school grounds between the bells.  It appears students would still be able to bring their smartphones onto school property, but Hochul’s bill would leave the matter of storing, collecting and distributing the phones to individual schools, who would be required to develop their own plans. Some school officials have expressed concerns their districts would not have the resources nor staff to collect, store and return hundreds of phones per day. They also raise questions about whether the policy would increase the risk of suspensions for students who don’t comply. Under the governor’s policy, there would be exceptions for students who require smartphones to manage medical needs, learning disabilities and for non-native English speakers. Hochul is including this proposal in her budget package, which has a deadline of April 1 to be passed by the State Legislature.