WI: Better Prussian performance against Napoleon in 1806

[GauchoBadger]

So, is there any way the early Fourth Coalition (an underrated part of the Napoleonic Wars, IMO), mainly Prussia and Saxony, could have fared better in its war against Napoleon in the year 1806?
Considering the Battles of Jena-Auerstedt, i've been under impression that Jena could not have resulted in a Prussian victory. The war, in fact, had the odds stacked against the Coalition, which was still using outdated tactics and command structures. Auerstedt, however, could have been a Prussian victory, considering the Prussian numerical advantage and Bernadotte's harmful reluctance to intervene in the field.
Assuming that this happens and that the Prussian army retreats in better shape after defeat at Jena, rather than scattering into multiple isolated armies, what next?
Will the French have to besiege Berlin, instead of occupying it with little resistance as happened IOTL?
What would the rematch battle be like?
Could Saxony's defection be prevented?

 

[unprincipled peter]

It's been debated that Spain was preparing to turn on Napoleon, and had started mobilizing. The shockingly rapid defeat of Prussia changed Godoy's mind. Spain had tried to walk a fine line of being France's ally while not really doing much to help. Nap was getting tired of this act, and Spain correctly saw that he would turn on them. The mobilization only cemented the deal.

Spain likely doesn't fare well against France, but it does give Britain a place to insert troops, and it does put France on a two front war. I'm guessing France still comes out on top. The butterflies of Portuguese crown moving/not moving to Brazil or how things shake out in the Spanish colonies might be significant.

 

[GauchoBadger]

One monkey wrench i notice in regards to an Anglo-Prussian alliance is the status of Hanover -- would the British be okay with a Prussian intent on annexing the home province of their monarch?