Well, here how I would do it (without getting in too much detail)

I would keep track of the value of the all the results I dropped. I would probably put them in a sorted queue whose size is the allowed number of dropped test. It would sorted such as the dropped test the nearest to zero is at the start. As I go trough the list with the traditional algorithm, I would do the following :

if I encounter a negative number
    if the queue is not full
        add it to the queue
    else
        if the new negative number is smaller (farther from zero) than the first number in the queue
            remove the first number from the queue
            add the new number to the queue
            add the removed number to the current subsequence value
        else
            add the new negative number to the current subsequence value

That way, you always keep the subsequence without the worst mark of the sequence, and if you encounter an even worst mark, you can restore the value of a previously dropped mark to the value of your subsequence.

The original algorithm is basically:

for each position:
    calculate best range ending at this position
print best over all possible ending positions

The best possible range ending at a position is the maximum of:

• 0 (starting from scratch here)
• the best possible range of the previous position + new mark

Now, we need to calculate K endings, for different amount of dropped tests

for each position:
    for k = 0 to K:
        calculate best possible range ending at this position and dropping at most k tests
print best of all calculated ranges

The best possible range ending is the maximum of:

• 0 (starting a new range at this position)
• the best possible range of the previous position + new mark
• the best possible range of the previous position and one less dropped test