Hello Jay,
First you need to know what template you are going to modify
https://github.com/odoo/odoo/blob/e2788b580ef15ef3083ac919737a46d850143832/addons/website_hr_recruitment/views/website_hr_recruitment_templates.xml#L131
look since 131 line - you got the "apply" template. Now your job is to inherit it, and tell with xpath what to do (add/remove/modify) so... to add a field there you first need to know the field name (if it exists) or create one if you need it.

Then tell with xpath to insert the additional block of your custom code including your field, respecting the template structure. To add a field after the phone number code will be like this:

<template id="user_navbar_inherit_website_hr_recruitment" inherit_id="website_hr_recruitment.apply">
    <xpath expr="//form[@id='hr_recruitment_form']/div/div[3]" position="after">
        <div class="form-group col-12 s_website_form_field s_website_form_required" data-type="char" data-name="Field">
            <div class="row s_col_no_resize s_col_no_bgcolor">
                <label class="col-form-label col-sm-auto s_website_form_label" style="width: 200px" for="recruitment8">
                    <span class="s_website_form_label_content">Your custom field</span>
                    <span class="s_website_form_mark"> *</span>
                </label>
                <div class="col-sm">
                    <input id="recruitment8" type="tel" class="form-control s_website_form_input" name="your_custom_field" required=""/>
                </div>
            </div>
        </div>
    </xpath>
</template>

PasteBin version: https://pastebin.com/kwjyNfsg

result link
https://ibb.co/0C9L0Wv

That's it about the form.

Regarding the data of the custom field .. please take a closer look at the right controller.
To find it first look where the Submit button will procees the form - URL (bottom left corner)
https://ibb.co/WN9J1PK

Now just find ther right controller check how it processes the data and modify by inheriting
controller is here:
https://github.com/odoo/odoo/blob/2e728f21bbc2771a8793d20e371f2fb7aea33e80/addons/website_hr_recruitment/controllers/main.py#L92

It's really to much to explain how to modify.. you need to be precise here. But in the weekend you can study a little bit.

Temporary, vote +1 if I helped. :)