You might describe such a person as 'Playing the Victim'.
The phrase can cover a range of behaviours, but includes this kind of 'poor, little me' manipulativeness. Wikipedia describes this:
Manipulators often play the victim role ("poor me") by portraying themselves as victims of circumstances or someone else's behavior in order to gain pity or sympathy or to evoke compassion and thereby get something from someone. Caring and conscientious people cannot stand to see anyone suffering, and the manipulator often finds it easy and rewarding to play on sympathy to get cooperation.
It appears that in psychology terms such a person may be described as a 'Dependent Addict'. This type is described in Human Manipulation - A Handbook
By Malcolm Coxall
seeks to inspire constant pity and a need to be helped, supported, guided and perhaps manipulated. In this way they will let others do their work for them.... The addict can be parasitic, eternally childish, a hypochondriac, a helpless individual or a whiner.
However, not every person who likes to get their own way can fairly be classed as suffering psychological disorder and the phrase 'dependent addict' may be prone to being misinterpreted as refering to substance addiction, so I'd advocate caution in deployment of that term. Colloquially, I might call such a person a 'professional victim'. Within my family we call such a person a 'helpless Hetty', but that doesn't seem to be a term the Internet knows.