Why there is a mark on my screen after showing the same thing for a long time? It is LCD so it's not the same problem as OLEDs I think.
The mark you are describing may be caused by a phenomenon known as image retention or burn-in, which can occur in LCD screens as well as OLEDs.

Image retention happens when a static image or a pattern is displayed on a screen for an extended period. The pixels responsible for showing that image may become stuck or frozen in that position, which can leave a faint ghost image of the previous display even when the screen changes to a new image.

To prevent image retention, you can try displaying a screensaver or turning off the display when it's not in use for an extended period. Additionally, newer LCD screens usually have built-in measures like pixel shifting or automatic brightness adjustment to mitigate this issue.

If the mark on your screen persists, it might be a more severe case of burn-in. In this case, you should consult with a professional or the manufacturer for possible solutions.
What are the potential options to repair such burn-in pixels? Is this something which is covered by manufetrurers guarantee?
Unfortunately, image persistence or screen burn-in on LCD screens is typically permanent and cannot be repaired. This is because LCD screens use a backlight to illuminate the pixels, and the backlight itself can cause uneven aging of the pixels over time, leading to permanent damage.

If your LCD screen is still under warranty, you may be able to get a replacement or repair from the manufacturer, depending on the specific terms of the warranty. However, most manufacturers' warranties do not cover damage caused by image persistence or screen burn-in, as this is considered a result of user misuse or prolonged exposure to static images.

To prevent further damage and prolong the life of your LCD screen, it is important to avoid displaying static images for extended periods of time and to use a screensaver or sleep mode when the screen is not in use. Additionally, you can try reducing the brightness and contrast settings on your screen to help minimize the risk of image persistence.