You might be able to appeal to the First-Tier or Upper Tribunal (Immigration and Asylum Chamber) against the Home Office decision. Your application can be made either in-country or out-of-country with a right for your appeal to be determined either on the papers, at an oral hearing, or at a substantive hearing before an Immigration Judge.
You must first be able to make a case for why the decision was legally wrong, for example, the Home Office failed to apply the correct law or wrongly interpreted the law, they failed to follow the correct procedures or they had no evidence or insufficient evidence to support its decision.
There are strict timeframes within which an appeal must be made to the relevant Tribunal, within 14 days if you are in-country or within 28 days if you are out-of-country. All the evidence you are relying on in your immigration matter will need to be lodged at the court including statements from any witnesses wishing to give evidence in support of your matter.