Can I leave for a certain period of time (say one month) and then have the right to re-enter and stay for another month (or three months if necessary)?
Basically, the answer is Yes. As you may know, this is subject to Directive 2004/38 / EC
There is not much clarity about it because it just is not enforced very regularly. In fact, an EU citizen can stay unlimited in another state only under very limited conditions. Under these conditions, the host country may require the registration of EU citizens. But any penalty for non-registration must be "proportional" to penalties for its own citizens who do not register. (That's why, strictly speaking, the UK has none need Registration.)
(I assume that the exceptions in terms of public safety, public health and public order are irrelevant.)
Accordingly, there is not much reason to hunt people who have not registered and have been present for more than three months. The benefits just do not justify the costs. Moreover, such enforcement efforts would be extremely difficult as the EU does not stamp the passports of EU citizens. It would not be easy to determine the location of the person and the actual length of stay. On the other hand, someone who had Crossing the border from one country to another and the theoretical reassessment of the three-month period may also have difficulty in providing evidence of this fact.
Consider someone who is They are informed by the authorities of a country that they have not complied with the registration obligation. Such a person will probably register or maybe just go. Or maybe the authorities accept the person's claim that she has left and returned. In this case, they could enforce a later registration deadline. Regardless, it would not be worthwhile for both parties to go to court. For this reason (as far as I know) there is no precedent to arrive at an answer.
Someone who has been present in an EU country other than your own for less than three months is not subject to registration, so your question could be reworded as follows:
What is required after a three-month stay in an EU country in the context of free movement to reset the three-month period?
The answer probably depends on several factors and above all on a specific analysis of each case. Imagine someone who pulls all his things into a house in the country, lives in that house and then goes away for a month. After the return of the person, it must be stated that the person has exceeded the three-month residence permit. Someone who travels the country for three months with a backpack might reasonably argue that he is resetting the period with an absence of just a day or two. In addition, different EU countries will consider this question differently (remember that the UK does not even require registration), so the answer will also depend on the country.
The above assumes that the person is self-sufficient and therefore falls under Article 7 (1) (b). If this is the case, the person can neither be deported from the country nor be refused entry. If this is not the case, then the person can be expelled from the country. Whether they can be denied later, seems to be controversial. The linked article states that the directive is very clear that it can not, and yet two governments are trying to create such an option (see also "Sleeping is not abusing EU free movement rights").
If someone who is not self-sufficient repeatedly enters a country and becomes a nuisance, a ban on "abuse of rights" (Article 35) can be imposed. However, it is very unlikely that a second entry, one month after a first three-month stay, will reach this level.