As a producer, having to work with a tight budget means you're looking for ways to save money, and preserve your net revenue from the project. You may be considering eliminating the soundman from your production team -- just have the cameraman, or a production assistant, take care of it. The sound will be good enough, and you'll save some money. Seems like a win-win for you, and for your client.
However, be sure you understand the downside risks: here are just a few possibilities:
This last example actually happened to a client of mine. The inexperienced, poorly trained (but inexpensive) production assistant they used for sound didn't understand that what he was listening to at the mixer wasn't what was being recorded on the camera. He had failed to set up the mixer to the camera properly, and recorded an entire day of interviews with the camera mic. The material was unusable and all the interviews had to be redone. This time they hired me to do the job. It was embarrassing to watch the producer try to explain to her client why all the interviews needed to be re-done, without saying what really happened. By trying to save money (the difference between a soundman's rate and a production assistant's rate), the producer lost the cost of an entire day of shooting, a full day of her time, and potentially her client's trust.
So, using a soundman on your team does two things. On the upside, it gives you the best quality sound under all the varying circumstances that you can encounter on location. And, it maximizes the quality of your images by freeing your cameraman to focus on their craft. On the downside, it prevents you from experiencing a significant loss of money and credibility by eliminating the possibility of ending up with unusable audio.