A map package or layer package can't be added directly into a web map. But your home user license includes an ArcGIS Online account and service credits that will let you publish services from ArcMap via ArcGIS Online's cloud, so if you can add your image into ArcMap, you can publish it as a service and add it to your web map. To check that you access to publishing services from ArcMap with your ArcGIS Online account, start ArcMap and choose File > Sign In with your ArcGIS Online account (if you are not already signed in). Now open any map document or add any data into your map. Make sure you are in Data view and not Layout view in ArcMap. Finally go to File > Share As Service and press Next in the wizard that appears and drop down the Choose a Connection list. If you can publish to ArcGIS Online you'll see a 'My Hosted Services' entry in that list. If you can't, you won't be able to. (If you have access to ArcGIS Server because you work for an organization that has ArcGIS Server, you can connect to that server at this point from the wizard to publish services).