Residents at the Jutland centre are not incarcerated but are required to live at the centre and register with authorities there up to seven times weekly.
But registrations have been missed 10,057 times since the centre was opened in 2016, Central and West Jutland Police have confirmed according to a report by newspaper Jyllands-Posten.
Some have absconded completely while others have absented themselves for long or short periods before returning, according to the report.
As such, a single individual can be responsible for large quantities of missed registrations.
243 people have been charged by police for failing to register, while other charges for failure to comply with rules relating to staying on the site have also been made.
The so-called udrejsecenter (departure or expulsion centre) houses rejected asylum seekers who have not yet left Danish territory, for example due to being stateless or because no readmission arrangement exists between Denmark and their home country.
Others do not agree with the outcome of their asylum application or are unable to be voluntarily returned to their home countries due to the danger they believe that will place them under.
The facility is home to 189 people at the time of writing.
In October 2017, residents at the centre protested at what they called 'intolerable' circumstances surrounding their status.
An expert told The Local that living conditions at the centre had been proven by studies to cause psychological damage.
Denmark's prosecution service has now designated a special prosecutor to deal with the missed registration cases, according to Jyllands-Posten, with two days since the turn of the year having seen cases at Herning City Court.
63 foreign citizens have reportedly been prosecuted over the issue, with fines and short prison sentences given to some while others have been acquitted.
Defence lawyer Kaare Kristensen criticised the legal process as futile, with the majority of residents having no means of earning money or paying back the fines given their status as non-residents.
Kristensen and fellow defence lawyer Jakob Fastrup both said that they had experienced clients being pleased to receive prison sentences. The individuals said that prison was a preferable place to be detained than the isolated expulsion centre, according to Jyllands-Posten's report.
Immigration minister Inger Støjberg told Jyllands-Posten that residents of the centre had "no business being in Denmark".
"Until we can get them out, we need to keep track of where they are staying, namely at Kærshovedgård. If they don't stay there, we''ll punish them and put them in prison," the minister added.