Very basic implemention of frontend logic required to display backups and create a new one

app/Http/Controllers/Api/Client/Servers/BackupController.php

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+<?php
+
+namespace Pterodactyl\Http\Controllers\Api\Client\Servers;
+
+use Pterodactyl\Models\Server;
+use Pterodactyl\Transformers\Api\Client\BackupTransformer;
+use Pterodactyl\Http\Controllers\Api\Client\ClientApiController;
+use Pterodactyl\Http\Requests\Api\Client\Servers\Backups\GetBackupsRequest;
+use Pterodactyl\Http\Requests\Api\Client\Servers\Backups\StoreBackupRequest;
+
+class BackupController extends ClientApiController
+{
+    public function __construct()
+    {
+        parent::__construct();
+    }
+
+    /**
+     * Returns all of the backups for a given server instance in a paginated
+     * result set.
+     *
+     * @param \Pterodactyl\Http\Requests\Api\Client\Servers\Backups\GetBackupsRequest $request
+     * @param \Pterodactyl\Models\Server $server
+     * @return array
+     */
+    public function index(GetBackupsRequest $request, Server $server)
+    {
+        return $this->fractal->collection($server->backups()->paginate(20))
+            ->transformWith($this->getTransformer(BackupTransformer::class))
+            ->toArray();
+    }
+
+    /**
+     * Starts the backup process for a server.
+     *
+     * @param \Pterodactyl\Http\Requests\Api\Client\Servers\Backups\StoreBackupRequest $request
+     * @param \Pterodactyl\Models\Server $server
+     */
+    public function store(StoreBackupRequest $request, Server $server)
+    {
+    }
+
+    public function view()
+    {
+    }
+
+    public function update()
+    {
+    }
+
+    public function delete()
+    {
+    }
+}

app/Http/Requests/Api/Client/Servers/Backups/GetBackupsRequest.php

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+<?php
+
+namespace Pterodactyl\Http\Requests\Api\Client\Servers\Backups;
+
+use Pterodactyl\Models\Permission;
+use Pterodactyl\Http\Requests\Api\Client\ClientApiRequest;
+
+class GetBackupsRequest extends ClientApiRequest
+{
+    /**
+     * @return string
+     */
+    public function permission()
+    {
+        return Permission::ACTION_BACKUP_READ;
+    }
+}

app/Http/Requests/Api/Client/Servers/Backups/StoreBackupRequest.php

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+<?php
+
+namespace Pterodactyl\Http\Requests\Api\Client\Servers\Backups;
+
+use Pterodactyl\Models\Permission;
+use Pterodactyl\Http\Requests\Api\Client\ClientApiRequest;
+
+class StoreBackupRequest extends ClientApiRequest
+{
+    /**
+     * @return string
+     */
+    public function permission()
+    {
+        return Permission::ACTION_BACKUP_CREATE;
+    }
+
+    /**
+     * @return array
+     */
+    public function rules(): array
+    {
+        return [
+            'name' => 'nullable|string|max:255',
+            'ignore' => 'nullable|string',
+        ];
+    }
+}