I did some analysis with the given example script. Related issue where Aaron has some analysis : issue12154 . The patch doesn't break anything since there are no tests for this that also could be added.

1. When we do inspect.getdoc(c.sum) it looks for obj.__doc__ (c.sum.__doc__) returning partialmethod.__doc__ ("new function with partial...") instead of checking if c.sum is a partial method and  returning obj.func.__doc__ which contains the relevant doc ("sum doc"). Thus getdoc needs to check before trying for obj.__doc__ if the obj is a partialmethod or partial object thus returning the original function object.

2. When we do inspect.getdoc(Child.sum) it looks for obj.__doc__ (Child.sum.__doc__) and since Child.sum is a partialmethod which has __get__ overridden it calls _make_unbound_method that returns a _method object with no doc and thus returning None. partialmethod object copies objects from the given function at https://github.com/python/cpython/blob/f1b9ad3d38c11676b45edcbf2369239bae436e56/Lib/functools.py#L368 and the actual object is returned at https://github.com/python/cpython/blob/f1b9ad3d38c11676b45edcbf2369239bae436e56/Lib/functools.py#L401 . Here self.func has the original function in this case Base.sum and _method._partialmethod has reference to Base.sum which contains the relevant docs but _method itself has no docs thus pydoc doesn't get any docs. So we can set _method.__doc__ = self.func.__doc__ and getdoc can pick up the docs.

sample script with

print(pydoc.render_doc(Child))
print(pydoc.render_doc(c))
print(inspect.getdoc(c.sum)) # Need to patch getdoc before it checks for obj.__doc__ to check if the c.sum is an instance of partialmethod or partial
print(inspect.getdoc(Child.sum)) # Need to patch functools.partialmethod._make_unbound_method to copy the docs to _method

Output :

```
Python Library Documentation: class Child in module __main__

class Child(builtins.object)
 |  Methods defined here:
 |
 |  diff(self, arg1, arg2)
 |      diff doc
 |
 |  sum = _method(self, arg2)
 |      sum doc
 |
 |  ----------------------------------------------------------------------
 |  Data descriptors defined here:
 |
 |  __dict__
 |      dictionary for instance variables (if defined)
 |
 |  __weakref__
 |      list of weak references to the object (if defined)

Python Library Documentation: Child in module __main__ object

class Child(builtins.object)
 |  Methods defined here:
 |
 |  diff(self, arg1, arg2)
 |      diff doc
 |
 |  sum = _method(self, arg2)
 |      sum doc
 |
 |  ----------------------------------------------------------------------
 |  Data descriptors defined here:
 |
 |  __dict__
 |      dictionary for instance variables (if defined)
 |
 |  __weakref__
 |      list of weak references to the object (if defined)

sum doc
sum doc

```

patch : 

diff --git a/Lib/functools.py b/Lib/functools.py
index ab7d71e126..751f67fcd0 100644
--- a/Lib/functools.py
+++ b/Lib/functools.py
@@ -398,6 +398,7 @@ class partialmethod(object):
             return self.func(*call_args, **call_keywords)
         _method.__isabstractmethod__ = self.__isabstractmethod__
         _method._partialmethod = self
+        _method.__doc__ = self.func.__doc__ or self.__doc__
         return _method

     def __get__(self, obj, cls):
diff --git a/Lib/inspect.py b/Lib/inspect.py
index b8a142232b..2c796546b2 100644
--- a/Lib/inspect.py
+++ b/Lib/inspect.py
@@ -600,6 +600,9 @@ def getdoc(object):
     All tabs are expanded to spaces.  To clean up docstrings that are
     indented to line up with blocks of code, any whitespace than can be
     uniformly removed from the second line onwards is removed."""
+    if isinstance(object, (functools.partialmethod, functools.partial)):
+        return object.func.__doc__
+
     try:
         doc = object.__doc__
     except AttributeError: