Rejection of a job offer in favor of another. Is it worth trying to fight back for the first one? I have a problem - I am looking for a job and I was on one of the job interviews, which due to the type of this job(electronics constructor/designer) it was the "target" choice that I dreamed about. The only drawback of the offer itself was that it is in a city far from the city where I come from.
Although I like this city anyway, the fact of moving and arranging my life completely from scratch has made me feel that I have to be a bit more careful with job searching, otherwise the possible failure in new job will cost me much more.

Unfortunately, it is not without significance that in my own region to find a similar job is almost impossible without experience - and my experience is only as hobbyist and and a little from the university (master degree).
In the meantime, however, after I agrred to join #1 workplace and waiting for a job letter with official employment offer, another company called me, but now in my neighborhood, to which I applied a long time ago and arranged a job interview the same day, 2 hours after. However, they offered me a completely different position in the office, related more to the supply of electrical parts.
On the one hand, at first, they offered financial conditions very similar to 1 # job (it was suspicious, because I used to be in the internship four years ago in #2 workplace as an electrician and I know that in general they paid very little to most employees), but this kind of job was not my dream and I still preferred to be in the electronics developing team - just like in # 1 job.

However, after all, after 2 days (although the boss from 2 # job insisted at the beginning, to let him know about the decision on the same day as the conversation), I chose the second job, which wasn't an easy decision. I still had remorse, but somehow I was explaining myself that it would somehow go away.
At the end I will write why I did not finally take the first offer.
But then something that disappointed me happened. The company boss # 2 arranged all terms of cooperation with me (including monthly wage) with me and directed me to make occupational medicine before work. On the second day I was to receive a referral for this research, but an hour before HR recruiter call me and says that the boss decided to reduce the monthly salary by about 20%, which made me (dutifully or not) angry. I quit this offer no. #2 finally, but and I know that if it was not a fraud and missing a promise of the company, I would choose the first offer (#1).

Why I rejected #1 job at first? I analyzed my fears, motivation, all pros and cons (both a few days ago and at the moment) and I know that:
1) #2 job boss pulled the wool over my eyes with potential opportunities, "fake" salary and why am I the most suitable person. First workplace employees were more careful and conservative when it comes to talking about work. Once, they even stated that their work is specific (after all, a work related in a way to the military industry). More infos I really learned from the profiles of other electronic designers from their linkedin profiles. the fact, however, that they called me for an interview a little later than they promised, in addition, they asked me for patience, because the recruitment is delayed in time.
I already sensed that they did not like someone like me with a miserable experience, but now I was sure they were waiting for somebody more experienced - and ultimately they might take me for lack of other possibilities. And that's how it initially happened - before I rejected their offer in favor of work no. # 2, I rejected # 1.

Now I see that I lost with my naivety and ability to persuade skill of the boss no. # 2. I lost also with my fear of the unknown, which in combination with moving so far from my home, probably also a lack of believing in my own abilities caused that I got scared of offer # 1, which I regret; c
It could never hurt to contact job #1. Maybe they will still need you. I would recommend that you avoid telling them that you are only applying to them because another job rejected you. People do not like being the fallback plan. Simply explain that with more time to consider your options, you came to realize that Job #1 is an excellent opportunity and that the move/distance from you current location should not be a barrier to pursing the important opportunity with #1. 

Understand, of course, that they may have already filled the position with someone else. In that case, all you can do is move on.

p.s. Never deal with people who renege on agreed to terms of employment. That should never be tolerated.