The short answer is no, you are not obligated to use what she provides. As long as you are providing for their basic needs (food, shelter, clothing) on your own then you are satisfying your parental obligations. The fact that the mother is attempting to get you to use whatever she provides is her trying to impose her will on your separate time with your children.
I would send the clothing back next time exactly as it was sent. If she brings it up, you can notify her that the children have adequate clothing at your home.
With regard to the cell phone, if you want to take it and turn it off, there is nothing wrong with that.
I hope this helps and thanks for the question.
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