The Working Time Regulations say that adult workers (aged 18 and over) who work for 6 hours or more must be given a break of at least 20 minutes and the break must be taken during the six hour period and not at the beginning or the end of it.
It is a requirement that employees be allowed to take their break away from their workstation, although not necessarily off the premises. Rest periods, including the statutory minimum of 20 minutes, can be unpaid as long as this is made clear in the Statement of Terms.
It sounds as though you are on duty without a break. I am assuming you are over 18, and if your shift lasts for 6 hours or more then you are entitled to a 20 minute break before the end of the shift. What you do with it is up to you as long as you don’t breach your employer’s policies and procedures (which may state that you have to remain on the premises for example).
However, you cannot be required to be on duty during these breaks as this defeats the object of the rest break! So when you ask can you claim additional payment, I would have to say not, but that rather you should be asserting your right to take the break from work in accordance with the Working Time Regulations.
See also: Are extra smoking breaks unfair to your employees?