const cheerio = require("cheerio");

const html = `<div class="contacts">
    <h3>Contacts</h3>
        <p>Website:&nbsp<a href="http://some.link/here" target="_blank">some.link</a></p>
        <p>Address:&nbsp<span>some address</span></p>
    </div>`;
const $ = cheerio.load(html);

const contacts = $("div.contacts p")
  .map((i, p) => $(p).text())
  .get();

console.log(contacts);

[
  'Website: some.link',
  'Address: some address'
]

[
  'Website:Áa href="http://some.link/here" target="_blank">some.link',
  'Address:Áspan>some address'
]