Unfortunately, the following problem exists:
A new external microphone is connected to the notebook via the integrated 3.5 jack connection and is also recognized as an external microphone by the Windows 10 operating system.
If you speak into the microphone, the voice is also recognized via Windows 10, but unfortunately no sound is output via the integrated notebook speakers.
So how can I use the external microphone in such a way that the voice can also be output via the integrated notebook speakers at the same time?
The notebook only has a single 3.5 jack connection.
That makes some sense because of the feedback. My suggestion is to use an audio interface via USB with which you use a system and speakers connected to it. In any case, the quality is then considerably better.
So how can I use the external microphone in such a way that the voice can also be output via the integrated notebook speakers at the same time?
This would require a direct analog signal path, otherwise there's inevitably at least a low latency (delay) due to the conversion from analog to digital to analog, which is immediately perceived as annoying, especially when speaking / singing.
Large buffers are used, especially when the sound chip is addressed via the "usual" APIs (e.g. WASAPI), because the most common desktop applications, such as music or video playback, do not require low latency. Applications that require low latency put the sound chip into a special mode and then address it using separate APIs that were specially developed for applications with low latencies, for example via ASIO or JACK.
Most external audio interfaces can easily do something similar to what you intend to do. Most support "direct monitoring", ie a direct analog signal path from the inputs to the outputs. In addition, external interfaces can often achieve lower latencies than the very inexpensive onboard audio chips, so that at least lower latencies are possible than is the case with onboard audio even without "direct monitoring". The delay should still be irritating when speaking / singing. In addition, other microphones and a "system" or headphones are needed because the audio outputs on the external interface can't be connected to the notebook speakers.
A Y adapter would probably work.