Hi Margole
As you are in receipt of CB ESA then your partners income will not be taken into account, even his state and private pensions. You will not need to contact DWP unless you are receiving a means tested benefit.
If you are only receiving Contribution Based ESA and you receive a pension lump sum, this will not affect your ESA but a regular payment will. If the pension is more then £85/week then there will be a £1/week deduction for every £2 in excess of the £85. You would then have to inform DWP of your pension.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
