Sure you can just get connected components (of certain size) with findContours or floodFill, and erase them leaving some smear. However, if you like to do it right you would think about why do you have the black area in the first place.
You did not use adaptive thresholding (locally adaptive) and this made your output sensitive to shading. Try not to get the black region in the first place by running something like this:
Mat img = imread("desk.jpg", 0);
Mat img2, dst;
pyrDown(img, img2);
adaptiveThreshold(255-img2, dst, 255, ADAPTIVE_THRESH_MEAN_C,
THRESH_BINARY, 9, 10); imwrite("adaptiveT.png", dst);
imshow("dst", dst);
waitKey(-1);
In the future, you may read something about adaptive thresholds and how to sample colors locally. I personally found it useful to sample binary colors orthogonally to the image gradient (that is on the both sides of it). This way the samples of white and black are of equal size which is a big deal since typically there are more background color which biases estimation. Using SWT and MSER may give you even more ideas about text segmentation.