You can use the Mock.Callback-method:
var mock = new Mock<Handler>();
SomeResponse result = null;
mock.Setup(h => h.AsyncHandle(It.IsAny<SomeResponse>()))
.Callback<SomeResponse>(r => result = r);
// do your test
new Foo(mock.Object).Bar(22);
Assert.NotNull(result);
If you only want to check something simple on the passed in argument, you also can do it directly:
mock.Setup(h => h.AsyncHandle(It.Is<SomeResponse>(response => response != null)));