What is the difference between MODEL_FILENAME_mode.mat and MODEL_FILENAME_mh_mode.mat?

I understand MODEL_FILENAME_mode.mat is output after set mode_compute>0.
And MODEL_FILENAME_mh_mode.mat is output (mode and covariance of mcmc draws) after metropolis hasting.

Am I rigth?

I am using option mode_file based in MODEL_FILENAME_mode.mat, by doing this I get “tigth” distributions (very very concentrated around some point), is this a bad signal?

I think the tightness of distributions is because my initial point of mcmc is based in MODEL_FILENAME_mode.mat which is the mode, and it should be change if I set mode_file based in mean of priors.

The MODEL_FILENAME_mode.mat is based on the results from mode finding, i.e. mode_compute>0. The resulting covariance matrix comes either out of the respective optimizer or is a Hessian at the mode computed via a finite difference approximation.

The MODEL_FILENAME_mh_mode.mat is computed based on the MCMC draws, i.e. if mh_replic>0
Based on MCMC draws the covariance matrix can be computed as the covariance between the draws. Ideally, the approximation at the mode should be a good approximation of the distribution and thus the results from both should be very similar. If they are not similar, there are a couple of reasons for this. For example, the mode-finding may have been unsuccessul of the MCMC may not have converged yet. Thus, it is hard to tell which one is preferrable.
An additional complication is that the covariance matrix at the mode from mode-finding is used as the proposal density for the MCMC and an almost 0 variance in one direction will then propagate to the MCMC. This may be what you are experiencing.
Which one to use depends very much on oyur goal. Most people report HPDIs based on the MCMC. In this case, it is important to verify convergence.

Using the prior mean to start the MCMC is generally not a good idea.