JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/fancyapps/[email protected]/dist/jquery.fancybox.min.css" />
<a class="theLink" data-fancybox data-src="#modal_pu" href="javascript:;" class="cart-price-btn">АААА</a>
<a class="theLink" data-fancybox data-src="#modal_pu" href="javascript:;" class="cart-price-btn">ББББ</a>
<a class="theLink" data-fancybox data-src="#modal_pu" href="javascript:;" class="cart-price-btn">ВВВВ</a>
<div class="form_head-style" style="display: none;" id="modal_pu">
<form class="mailsend ajax-contact-form" action="путь к обработчику" method="POST">
<input type="text" name="Имя" value="" placeholder="Имя" required type="" required >
<input type="tel" name="Номер" value="" placeholder="Телефон" required>
<input type="email" name="email" value="" placeholder="E-mail">
<input type="submit" name="submit" value="Отправить" id="submit">
</form>
</div>
<script src="https://cdn.jsdelivr.net/gh/fancyapps/[email protected]/dist/jquery.fancybox.min.js"></script>

JavaScript

$('.theLink').on("click", e =>{
  addField(e)
})

function addField (e) {
	if ($('#hiddenInp').length==0){
  	$('.mailsend').append(`<input type="text" id="hiddenInp" name="Имя1" value="${e.target.textContent}">`)
  }
	
}