If I had to guess it is the server like Gonner said and the resources are being bogged down, but it could also be the theme requesting weird stuff on occasion and also your cache. Lets try this: Check cache plugins, including server or host side caching and disable. Go to plugin updates and reinstall the current version of wordpress which cleans the database tremendously. Try a new cache plugin and use
https://wordpress.org/plugins/wp-widget-cache/ which will reduce the load on your server significantly. Then please report back.