Mayor Eric Adams on Friday struck a handshake deal with the City Council on a record-breaking nearly $116 billion budget — which would create a new department to regulate e-bikes, sources said. The tentative agreement includes $6.1 million for a new “Department of Sustainable Delivery,’’ which would be staffed by 60 people and dedicated to e-bike regulation and enforcement after a recent push by the Adams administration to rein in the two-wheelers. City Hall also agreed to give the council a major boost in funding for immigration services to the tune of $54.5 million, a source familiar with the deal...