Regression model for a count proces
I have to quote Tukey, perhaps the grandfather of data science:


  The combination of some data and an aching desire for an answer does
  not ensure that a reasonable answer can be extracted from a given body
  of data.


I see nothing wrong with your Poisson model. In fact its a pretty good fit to the data. The data is noisy. There is nothing you can do about it. Perhaps the noise if due to whatever else is on TV at the time, or the weather, or the phase of the moon. Whatever it is, its not in your data.

If you reasonably think the weather might be affecting your data, get the weather data and add it. If it decreases the log-likelihood enough for each degree of freedom then it's doing a good job and you leave it in. This is regression modelling 101. 

Of course there's a zillion other things you can do. Scale the data by any old transformation you want. Fit a quadratic. A quartic. A quintic. A spline.  You could include the date and possible temporal correlation effects. But always bear in mind what Tukey was saying - if your data is noisy, you won't get anything much out of it. So it goes.