Greedily partition the list of combinations with repetition
Jelly, 26 25 bytes

œ&amp;µL‘&lt;⁴ȧ⁹ȯ
œċµç\L€=⁴œṗµḊ’


Full program which prints a representation of a list of lists, each list being a part, e.g. for n=5, k=3:

[[[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 0, 3], [0, 0, 4]], [[0, 1, 1], [0, 1, 2], [0, 1, 3], [0, 1, 4]], [[0, 2, 2], [0, 2, 3], [0, 2, 4]], [[0, 3, 3], [0, 3, 4]], [0, 4, 4], [[1, 1, 1], [1, 1, 2], [1, 1, 3], [1, 1, 4]], [[1, 2, 2], [1, 2, 3], [1, 2, 4]], [[1, 3, 3], [1, 3, 4]], [1, 4, 4], [[2, 2, 2], [2, 2, 3], [2, 2, 4]], [[2, 3, 3], [2, 3, 4]], [2, 4, 4], [[3, 3, 3], [3, 3, 4]], [[3, 4, 4], [4, 4, 4]]]


Note: the representation used removes the redundant [ and ] around lists of length 1.

Try it online! or see a pretty print version (cost 3 bytes)

How?

œ&amp;µL‘&lt;⁴ȧ⁹ȯ - Link 1, conditional multi-set intersection: list x, list y
œ&amp;         - multi-set intersection(x, y)
  µ        - monadic chain separation (call that i)
   L       - length(i)
    ‘      - increment
     &lt;     - less than?:
      ⁴    -     2nd program input, k
       ȧ   - logical and with:
        ⁹  -     link's right argument, y (y if i is too short, else 0)
         ȯ - logical or (y if i is too short, else i)

œċµç\L€=⁴œṗµḊ’ - Main link: n, k
œċ             - combinations with replacement(n, k) (sorted since n implies [1,n])
  µ            - monadic chain separation (call that w)
         œṗ    - partition w at truthy indexes of:
   ç\          -     reduce w with last link (1) as a dyad
     L€        -     length of €ach
        ⁴      -     2nd program input, k
       =       -     equal (vectorises)
           µ   - monadic chain separation
            Ḋ  - dequeue (since the result will always start with an empty list)
             ’ - decrement (vectorises) (since the Natural numbers were used by œċ)