What is the most appropriate English word to describe '面' in phrase like '看在A的面上, B...' I think the form is the same with '份' in '看在A的份上, B...' which had been asked before. 

However, I'm not sure 'for one's sake', which is the preferable answer for '份', is the appropriate word to describe '面'.

For example: '看在你妈妈的面上，我就不为难你'. In this case, I assumed the speaker has a kind of grateful feeling towards the mother and thus wouldn't make things difficult for the child.

Is there any english word to describe it more precisely than 'for one's sake'?
'看在A的面上, B...' is better be translated as " Out of respect to A, B..."
  
  Example: "看在你妈妈的面上，我就不为难你" - " Out of respect to your mother, I'll let you off the hook'


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  As for '看在A的份上, B...', I would translate it as 
  
  "For A's sake, B..." 
  
  or 
  
  "With consideration for A, B..."
  
  Examples: 
  
  "看在你妈妈的份上，我就不为难你" - "For your mother's sake, I'll let you off the hook" 
  
  or 
  
  "看在你妈妈的份上，我就不为难你" - "With consideration for your mother, I'll let you off the hook"