Within tyre manufacturing, a reinforcing structure of the tyre, such as the zero-degree layer of the belt structure, is made by an apparatus having devices for guiding a continuous elongated element towards a forming drum, devices for applying the elongated element against a laying surface of the drum, and a cutting unit to cut off the continuous elongated element when application has been completed. Once the cutting operation has been carried out, the free end of the continuous elongated element is set to a grip position downstream of the cutting unit, through a reciprocating slider and a threading element blocking the continuous elongated element with respect to the slider when the latter moves towards the forming drum.