This code snippet adds a link "Edit private content" to the profile under the private content.

um_profile_private_content_edit.php

<?php

/**
 * Add a button to the member profile that an admin can click so they are redirected to the private content of the relevant member
 * @author Ultimate Member support <support@ultimatemember.com>
 * @since  2022-03-17
 * @see    #59138
 */
add_action( 'um_profile_content_private_content', function( $args ) {
	if ( is_user_logged_in() && current_user_can( 'administrator' ) && current_user_can( 'edit_user', um_profile_id() ) ) {
		$private_content_link = UM()->Private_Content()->get_private_content_post_link( um_profile_id() );
		echo '<div class="um-col-alt"><a href="' . esc_url( $private_content_link ) . '" class="um-button">' . __( 'Edit private content', 'um-private-content' ) . '</a></div>';
	}
}, 20 );
add_filter( 'um_user_profile_tabs', function( $tabs ) {
	if ( empty( $tabs['private_content'] ) && is_user_logged_in() && current_user_can( 'administrator' ) && current_user_can( 'edit_user', um_profile_id() ) ) {
		$tab_title               = UM()->options()->get( 'tab_private_content_title' );
		$tab_icon                = UM()->options()->get( 'tab_private_content_icon' );
		$tabs['private_content'] = array(
			'name'            => ! empty( $tab_title ) ? $tab_title : __( 'Private Content', 'um-private-content' ),
			'icon'            => ! empty( $tab_icon ) ? $tab_icon : 'um-faicon-eye-slash',
			'default_privacy' => 2,
		);
	}
	return $tabs;
}, 1001, 1 );

Expected result: