Adjust Push Notification Logic

Nucleic-Session: 0F065CD6-20FE-4941-ACF5-C57A4572A5A3
Co-authored-by: Nucleic <[email protected]>

NucleicRemote/NucleicRemote/Info.plist

@@ -20,6 +20,14 @@
 	     target's "Stamp git commit into Info.plist" build phase (git rev-parse --short HEAD). -->
 	<key>NucleicCommit</key>
 	<string></string>
+	<!-- Wake the app in the background for the relay's silent "clear" push, which recalls the
+	     "waiting for your approval" tickle once the approval is resolved on another device
+	     (docs/CLOUD_INFRA.md §3). Without this mode iOS drops the content-available push and the
+	     stale notification lingers until the next launch. -->
+	<key>UIBackgroundModes</key>
+	<array>
+		<string>remote-notification</string>
+	</array>
 	<key>NSSupportsLiveActivities</key>
 	<true/>
 	<!-- Let the Macs' APNs pushes refresh the glance promptly while the phone is locked — without