diff --git a/include/pybind11/cast.h b/include/pybind11/cast.h
index cdeb4aa8b..fbccf8bf5 100644
--- a/include/pybind11/cast.h
+++ b/include/pybind11/cast.h
@@ -872,9 +872,8 @@ template <typename base, typename deleter>
 struct is_holder_type<base, std::unique_ptr<base, deleter>> : std::true_type {};
 
 template <typename T>
-struct handle_type_name {
-    static constexpr auto name = const_name<T>();
-};
+struct handle_type_name;
+
 template <>
 struct handle_type_name<object> {
     static constexpr auto name = const_name("object");
@@ -968,6 +967,18 @@ struct handle_type_name<type> {
     static constexpr auto name = const_name("type");
 };
 template <>
+struct handle_type_name<capsule> {
+    static constexpr auto name = const_name("capsule");
+};
+template <>
+struct handle_type_name<ellipsis> {
+    static constexpr auto name = const_name("ellipsis");
+};
+template <>
+struct handle_type_name<weakref> {
+    static constexpr auto name = const_name("weakref");
+};
+template <>
 struct handle_type_name<args> {
     static constexpr auto name = const_name("*args");
 };
@@ -975,6 +986,30 @@ template <>
 struct handle_type_name<kwargs> {
     static constexpr auto name = const_name("**kwargs");
 };
+template <>
+struct handle_type_name<obj_attr_accessor> {
+    static constexpr auto name = const_name<obj_attr_accessor>();
+};
+template <>
+struct handle_type_name<str_attr_accessor> {
+    static constexpr auto name = const_name<str_attr_accessor>();
+};
+template <>
+struct handle_type_name<item_accessor> {
+    static constexpr auto name = const_name<item_accessor>();
+};
+template <>
+struct handle_type_name<sequence_accessor> {
+    static constexpr auto name = const_name<sequence_accessor>();
+};
+template <>
+struct handle_type_name<list_accessor> {
+    static constexpr auto name = const_name<list_accessor>();
+};
+template <>
+struct handle_type_name<tuple_accessor> {
+    static constexpr auto name = const_name<tuple_accessor>();
+};
 
 template <typename type>
 struct pyobject_caster {
diff --git a/include/pybind11/pybind11.h b/include/pybind11/pybind11.h
index bc2f5924a..429d2138d 100644
--- a/include/pybind11/pybind11.h
+++ b/include/pybind11/pybind11.h
@@ -1190,6 +1190,15 @@ protected:
     }
 };
 
+PYBIND11_NAMESPACE_BEGIN(detail)
+
+template <>
+struct handle_type_name<cpp_function> {
+    static constexpr auto name = const_name("Callable");
+};
+
+PYBIND11_NAMESPACE_END(detail)
+
 /// Wrapper for Python extension modules
 class module_ : public object {
 public:
@@ -2618,6 +2627,11 @@ public:
 
 PYBIND11_NAMESPACE_BEGIN(detail)
 
+template <>
+struct handle_type_name<exception<void>> {
+    static constexpr auto name = const_name("Exception");
+};
+
 // Helper function for register_exception and register_local_exception
 template <typename CppException>
 exception<CppException> &
diff --git a/include/pybind11/pytypes.h b/include/pybind11/pytypes.h
index e7c7c8124..d5f6af8e0 100644
--- a/include/pybind11/pytypes.h
+++ b/include/pybind11/pytypes.h
@@ -59,6 +59,7 @@ struct sequence_item;
 struct list_item;
 struct tuple_item;
 } // namespace accessor_policies
+// PLEASE KEEP handle_type_name SPECIALIZATIONS IN SYNC.
 using obj_attr_accessor = accessor<accessor_policies::obj_attr>;
 using str_attr_accessor = accessor<accessor_policies::str_attr>;
 using item_accessor = accessor<accessor_policies::generic_item>;
diff --git a/tests/test_exceptions.py b/tests/test_exceptions.py
index a0c51c564..378ec6348 100644
--- a/tests/test_exceptions.py
+++ b/tests/test_exceptions.py
@@ -424,4 +424,4 @@ def test_fn_cast_int_exception():
 def test_return_exception_void():
     with pytest.raises(TypeError) as excinfo:
         m.return_exception_void()
-    assert "`exception<void>`" in str(excinfo.value)
+    assert "Exception" in str(excinfo.value)
diff --git a/tests/test_pytypes.cpp b/tests/test_pytypes.cpp
index 0cd480ebb..03d46ae98 100644
--- a/tests/test_pytypes.cpp
+++ b/tests/test_pytypes.cpp
@@ -41,6 +41,15 @@ class float_ : public py::object {
 };
 } // namespace external
 
+namespace pybind11 {
+namespace detail {
+template <>
+struct handle_type_name<external::float_> {
+    static constexpr auto name = const_name("float");
+};
+} // namespace detail
+} // namespace pybind11
+
 namespace implicit_conversion_from_0_to_handle {
 // Uncomment to trigger compiler error. Note: Before PR #4008 this used to compile successfully.
 // void expected_to_trigger_compiler_error() { py::handle(0); }