{!! Form::open(['action'=>'AdminPromotionsController@store', 'method' => 'POST','enctype'=>'multipart/form-data']) !!}
<div class="form-group">
     <form name="add_name" id="add_name">
          <div class="table-responsive">
               <table class="table table-bordered" id="dynamic_field">
                    <tr>
                         <td> {{Form::file('promotion_image[]')}}</td>

                         <td>{{ Form::button('Add', ['class' => 'btn btn-success', 'id'=>'add','name'=>'add']) }}</td>
                    </tr>
               </table>
               {{Form::submit('submit', ['class'=>'btn btn-primary'])}}
          </div>
     </form>
</div>
{!! Form::close() !!}

$this->validate($request, [
        'promotion_image' => 'required'
    ]);

   //Handle File Upload
   if($request->hasFile('promotion_image[]')){
    // Get FileName
    $filenameWithExt = $request->file('promotion_image[]')->getClientOriginalName();
    //Get just filename
    $filename = pathinfo( $filenameWithExt, PATHINFO_FILENAME);
    //Get just extension
    $extension = $request->file('promotion_image[]')->getClientOriginalExtension();
    //Filename to Store
    $fileNameToStore = $filename.'_'.time().'.'.$extension;
    //Upload Image
    $path = $request->file('promotion_image[]')->storeAs('public/promotion_images',$fileNameToStore);
    }else{
        $fileNameToStore='noimage.jpg';
    }

    $promotion = new Promotion;
    $promotion->promotion_image = $fileNameToStore;
    $promotion->save();

<script>
$(document).ready(function(){
     var i=1;
     $('#add').click(function(){
          i++;
          $('#dynamic_field').append('<tr id="row'+i+'"><td>{{Form::file('promotion_image[]',['class'=>'form-control name_list'])}}</td><td><button type="button" name="remove" id="'+i+'" class="btn btn-danger btn_remove">X</button></td></tr>');
     });
     $(document).on('click', '.btn_remove', function(){
          var button_id = $(this).attr("id");
          $('#row'+button_id+'').remove();
     });
     $('#submit').click(function(){
          $.ajax({
               url:"name.php",
               method:"POST",
               data:$('#add_name').serialize(),
               success:function(data)
               {
                    alert(data);
                    $('#add_name')[0].reset();
               }
          });
     });
});
</script>

$('#add_name').serialize() using this you can't get image object in post using ajax.

var form = $('form')[0];
var formData = new FormData(form);