@c67 

Which wireless band you are connecting to, 2.4G or 5G?

Have you tested the Cisco connection on other client devices? Do they behave the same?

Try to change the security to WPA2-PSK, AES for the main network, will it help?

  @c67,

Apologize for the late response. 

The Smart Connect Feature moves the responsibility of switching between the 2.4 and 5Ghz bands to the router, this means the router will automatically move your device to the band that has the strongest signal quality. If you disable the feature and set the SSID to be the same for both bands, the responsibility to switch between bands will move to the client device instead.

Have you tried setting a Static IP and IP reservation for the device? You may also want to try changing your channel width or channel to see if there is interference causing the disconnect. Check your QoS settings, you may want to manually set this to be above what you expect to receive from your ISP. 

Lastly, are you able to see (either in your clients connection status or your router logs) what may have caused the device to disconnect from the network or if the device is disconnecting from the network or from just the internet?

Going through some of the forums for AnyConnect, it looks like there may be ports that you should open that may be causing the disconnects: Cisco Anyconnect VPN Client Ports