Dear member,
The senior member KK!HR has already given a proper reply to your post. From your post, it appears that you are from a government organisation or a PSU.
When the employment is accepted, irrespective of the marital status, the employee is expected to accept the challenges that come in the wake of employment. Generally, the decision about transfers are taken at the Head Office and the decision-making authorities give weight to the administrative requirements rather than the employee's marital status. Against this backdrop, to honour one of the employee's requests to put off the transfer, they may have to redo the entire transfer order itself. Perhaps the senior authorities might not be interested in rework. Hence their disinclination.
The second reason why the senior authorities did not yield to your request could be that for them receiving such requests is common. If not the single motherhood, then there could be some other reason like the sickness of either parent or something else. On a rotation basis, all the employees have to be given a rural posting and they cannot single out any employee because of family issues.
How are your relations with your superiors? Does the transfer have a bearing on the relations? Does your salary correspond to the work that you do? By transferring you, do the seniors wish to reduce the employee cost of your branch?
Of course, all the is purely my guesswork and the facts could be different.
Since your application to stop the transfer has not been acceded to, you may escalate the matter and put up one more application addressed to the regional office or the zonal office. Let us see whether or not they agree.
Thanks,
Dinesh Divekar