[BUG] Adding offset with nonzero month to DatetimeIndex (GH26258)
State: closed

- [x] closes #26258 
- [x] 1 test added
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry


Labels:
name: Bug
name: Datetime
description: Datetime data dtype
name: Frequency
description: DateOffsets

Comments:
# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=h1) Report
> Merging [#26292](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/71207255f32af73cd701fe4ecf56a643c925a9e1?src=pr&el=desc) will **decrease** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/26292/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #26292      +/-   ##
==========================================
- Coverage   91.98%   91.98%   -0.01%     
==========================================
  Files         175      175              
  Lines       52374    52374              
==========================================
- Hits        48178    48175       -3     
- Misses       4196     4199       +3
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.53% <100%> (ø)` | :arrow_up: |
| #single | `40.71% <0%> (-0.16%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/tseries/offsets.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL3RzZXJpZXMvb2Zmc2V0cy5weQ==) | `96.69% <100%> (ø)` | :arrow_up: |
| [pandas/io/gbq.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2dicS5weQ==) | `78.94% <0%> (-10.53%)` | :arrow_down: |
| [pandas/core/frame.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZnJhbWUucHk=) | `96.9% <0%> (-0.12%)` | :arrow_down: |
| [pandas/util/testing.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL3V0aWwvdGVzdGluZy5weQ==) | `90.71% <0%> (+0.1%)` | :arrow_up: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=footer). Last update [7120725...04d5351](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=h1) Report
> Merging [#26292](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/6d7ba051508b93fbbc80f6929ed6f2254ab7b938?src=pr&el=desc) will **decrease** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/26292/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #26292      +/-   ##
==========================================
- Coverage   92.04%   92.03%   -0.01%     
==========================================
  Files         175      175              
  Lines       52291    52291              
==========================================
- Hits        48131    48127       -4     
- Misses       4160     4164       +4
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.59% <100%> (ø)` | :arrow_up: |
| #single | `40.72% <0%> (-0.15%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/tseries/offsets.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL3RzZXJpZXMvb2Zmc2V0cy5weQ==) | `96.69% <100%> (ø)` | :arrow_up: |
| [pandas/io/gbq.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2dicS5weQ==) | `78.94% <0%> (-10.53%)` | :arrow_down: |
| [pandas/core/frame.py](https://codecov.io/gh/pandas-dev/pandas/pull/26292/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZnJhbWUucHk=) | `97.01% <0%> (-0.12%)` | :arrow_down: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=footer). Last update [6d7ba05...e6da6bd](https://codecov.io/gh/pandas-dev/pandas/pull/26292?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

@jreback Do you have anything else to implement? Please tell me.
thanks @makbigc keep em coming!