woocommerce_add_cart_item_data.php

<?php
/**
 * Add custom cart item data
 */
function plugin_republic_add_cart_item_data( $cart_item_data, $product_id, $variation_id ) {
  if( isset( $_POST['pr-field'] ) ) {
    $cart_item_data['pr_field'] = sanitize_text_field( $_POST['pr-field'] );
  }
  return $cart_item_data;
}
add_filter( 'woocommerce_add_cart_item_data', 'plugin_republic_add_cart_item_data', 10, 3 );