Instead of trying to make this a Partial Class, you should just make a Property containing an instance of that class.
Nested classes must be created and have instances, just like top level classes. By making a property within your main class, you can automatically create that instance in your main class constructor, and your code will work as expected.
Public Class clsMain
Public Property myIntProp as Integer
Public property myStrProp as String
'other properties
Public Property OtherFunctionality as MyOtherClass = New MyOtherClass()
Then just define the class in a separate file:
Public Class MyOtherClass
Public Sub MyFunction1()