您的查询是获取至少有一个总部（headquarter_type设置为physical）的所有城市
如果你只想有城市只有物理总部，那么你应该使用这个：

$city_headquarters = City::whereDoesntHave('headquarters', function($query){
   $query->where('headquarter_type', '<>', 'physical');  
})->get();

如果只想立即加载物理总部，则：

$city_headquarters = City::whereHas('headquarters', function($query) {
   $query->where('headquarter_type','physical');
})->with([
    'headquarters' => function($query) {
          $query->where('headquarter_type', 'physical');
    }
])->get();

然后您可以：

foreach($city_headquarters as $city) {
   foreach($city->headquarters as $physical_headquarter) {
      $physical_headquarter->[...]
   }
}