The answer seems to be the last of your hypotheses. These "allowances" were deducted from the tenth and fifteenth tax liability: they were a rebate, selectively applied to towns deemed to be in need. This means, in effect, that some proportion of the tax which the towns in question would normally have paid was deducted. I am not completely sure whether this rebate or allowance was simply not collected from those towns in the first place, or whether the whole tax liability was collected and then the allowance returned to the town. I think it was probably the former.

The background to this is the long-standing tenth and fifteenth direct tax system, which was formalised in 1334, such that, according to established valuations, one whole tenth and fifteenth was held to yield a national total of £39,000:

"Hence, from 1334 onward, the following method of apportionment was employed, i.e. a tenth and fifteenth was taken as affording a definite sum measured by the yield on the ancient valuation. As this gave between £38,000 and £39,000 in the aggregate, the tenth and fifteenth became for the future practically a fiscal expression for a sum of about £39,000, the total to be divided or apportioned between the several counties, cities and boroughs according to their former payments."
http://en.wikipedia.org/wiki/History_of_the_English_fiscal_s...

How much in total each town and county had to pay was established by tax commissioners.

Therefore the total allowances of £4000 and later £6000 per whole tenth and fifteenth, granted in the period from 1433, amounted to a rebate of just over 10% and later just over 15% of the tenth and fifteenth tax. This sum was deducted from the total:

"The standard tax, the tenth and fifteenth, was failing to meet changing conditions in that, as towns decayed, further allowances had to be given amounting to over 15% (£6,000), which, with other deductions, lowered the yield from tenths and fifteenths to £31,000"
http://en.wikipedia.org/wiki/History_of_the_English_fiscal_s...

This certainly seems to indicate that the allowance of £6000 meant that £6000 of the tenth and fifteenth tax was not collected, or else it was collected and then returned.

To see how the allowances worked in practice, we need local information. The folllowing is from a study of medieval tax assessments in Essex.

First, it is interesting to note that allowances for the relief of local need had sometimes been granted from other sources, such as proceeds from fines:

"Although the basic village quotas of 1334 remained unaltered there were occasions when they were temporarily modified, abatements or reliefs being allowed in view of the impoverishment of a particular vill. Such occasions were the three collections of a tenth and a fifteenth granted in 1351 when impoverished vills were reimbursed out of a fund provided by fines collected under the Statute of Labourers."

However, these fixed allowances established in the fifteenth century were, as I have said, set against the tenth and fifteenth tax. The allowance to be given to each town seems to be been locally assessed, at least at first, and to have been granted in the form of a reduction of the town's tax burden:

"In 1433 there began a long series of abatements whereby a sum of £4,000 and later £6,000 was distributed among the over-taxed and impoverished villages of the kingdom. For at least the first 30 years of the abatements the evidence indicates that a genuine reassessment of need was made at each new collection of a subsidy; the Devonshire figures show quite wide differences in the sums allowed to each borough from one collection to the next. Unfortunately there are only two surviving rolls for Essex in this period, dated 1433 and 1436, and in these rolls the rate of allowances in Ongar hundred is the same in each year; the county was relieved of its obligation to pay £123 7s. 5d. (or about 10 per cent. of the sum due) and in its turn the hundred of Ongar was relieved in the same proportion, £6 12s. 6¾d. being allowed."
http://www.british-history.ac.uk/report.aspx?compid=15722

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Note added at 11 hrs (2012-02-06 07:43:55 GMT)
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Separately from this, other allowances ("reliefs" or "abatements") were sometimes paid, in particular circumstances, not (apparently) in the form of a rebate on tax payable but as a direct grant. The Essex study goes on to mention an example:

"In addition to the general abatements granted by statute, some villages in Essex seem to have been allowed a second sum for losses suffered per inundacionem aquarum et alia infortuna [sic] pericula. Thus Langham received 10s. for flood damage. No flood relief was given in Ongar, but Chipping Ongar obtained 3s. 4d. extra relief in 1436 for pericula infortuna which were not specified. The abatements were assessed by the Abbot of Colchester and the two knights of the shire in the current Parliament: in 1436 Edward Tyrell and Thomas Torell. A document from Totnes suggests that at the end of the Parliament the knights actually brought the relief back with them for distribution, but the procedure in Essex is not specifically known."