It's the office that administers the Court
I have retained the formality here, but effectively it's just the "court office"
or the "court administrator's office" to coin UK terms

TSJ Cataluña: "The Superior Tribunal of Justice culminates the judicial organization in the autonomous community with respect to the Supreme Tribunal's corresponding jurisdiction, on the matters which insist on constitutional guarantees and competencies which correspond to the constitutional tribunal."
Wikipedia: "In the UK, the Supreme Court is the final court of appeal for civil and criminal cases. It can consider cases of constitutional importance which affect the whole population, including disputes relating to devolution."

(SP: "El Tribunal Supremo de Justicia culmina la organización judicial en el ámbito territorial de la Comunidad Autónoma sin perjuicio de la jurisdicción que corresponde al Tribunal Supremo y de aquellas materias que exijan garantías constitucionales, competencia que corresponde al Tribunal Constitucional."
En Inglaterra la Corte Suprema es la corte final de apelación de casos civiles y criminales. Puede considerar casos de importancia constitucional que afectan la población entera, hasta disputas relacionadas con la devolución.")

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Note added at 11 hrs (2021-06-16 23:37:22 GMT)
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It's a tribunal with regional authorities alongside a governing organization. The High Court functions on a national level, not the same as a tribunal.
The disagreeing entity would gain a little credibility by providing references in support of its disagreements and also some answers.
Una 'secretaria de gobierno' is a government secretary, ie/ a secretary belonging to a government or a governing organization.
An administration is an administrative authority, not a government.

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Note added at 3 days 21 hrs (2021-06-20 10:22:09 GMT)
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https://www.judiciary.uk/about-the-judiciary/who-are-the-jud...

"The Upper Tribunal, primarily, but not exclusively, reviews and decides appeals arising from the First Tier Tribunal. Like the High Court, it's a superior court of record, as well as having the existing specialist judges of the Senior Tribunals Judiciary at its disposal, it can also call on the reserves of High Court Judges."

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Note added at 8 days (2021-06-24 23:50:07 GMT) Post-grading
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The Upper Tribunal takes place following decisions from a First Tier Tribunal, after a claimant has requested for his or her case to be re-considered, while he or she hopes for a better resolution.

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Note added at 8 days (2021-06-24 23:57:59 GMT) Post-grading
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SP: "El Tribunal de alto nivel, primariamente, aunque no exclusivamente, reconsidera y toma decisiones sobre el Tribunal inicial. Como el Tribunal Supremo, es un tribunal superior y registrado que tiene jueces especialistas, mientras puede llamar a magistrados de reserva de las Cortes."
El Tribunal de alto nivel se convoca a partir de las decisiones tomadas por el tribunal inicial, si la persona que reivindica, reclama o tiene algo que pedir, insiste en que sea reconsiderado su caso, mientras tiene la expectativa de que se resuelva mejor tras una segunda audiencia.