You can also use StartActivityForResult and use the result of the second activity for delete item in first one.
I mean:
- FirstActivity starts SecondActivity waiting for result
- SecondActivity sends the result back to FirstActivity. Only if you delete
the item.
- Now FirstActivity remove and refresh the list.
In FirstActivity:
Intent i = new Intent(this, SecondActivity.class);
startActivityForResult(i, 1);
In SecondActivity, when you push delete button:
Intent returnIntent = new Intent();
returnIntent.putExtra("delete", true);
returnIntent.putExtra("position", position);
setResult(Activity.RESULT_OK, returnIntent);
finish();
And finally, FirstActivity handle the result:
@Override
protected void onActivityResult(int requestCode, int resultCode, Intent data) {
    if (requestCode == 1) {
        if(resultCode == Activity.RESULT_OK){
            if (data.getBooleanExtra("delete") {
                 // get position and delete item from list and refresh
                 int position = data.getIntegerExtra("position");
            }
        }
        if (resultCode == Activity.RESULT_CANCELED) {
            //Write your code if there's no result
        }
    }
}//onActivityResult
https://stackoverflow.com/a/10407371/1820599 
Edited:
Getting the context of your activity inside the adapter constructor:
FirstActivity listener;
public myAdapter(Context context, List<String> items) {
        super(context, R.layout.row_edition, items);
        this.listener = ((FirstActivity) context);
        this.items = items;
    }
Then, inside the adapter, when you push on item, call the activity to start the seconde one:
listener.startSecondActivity(int position, parameters you need to use);
and finally, in your FirstActivity
startSecondActivity(int position, parameters you need to use) {
    // whatever you have to do
    Intent i = new Intent(this, SecondActivity.class);
    // push position inside intent and whatever you need
    startActivityForResult(i, 1);
}
The flow is: 
- Push item
- Use FirstActivityListener to call SecondActivity
- In SecondActivity delete and senr result back
- In FirstActivity remove item from adapter, using an auxiliar method
inside que adapter