Garrow, Alan J.P. "Chapter Eight: The Gospel Layer: 8.2b; 11.3b; 15.3-4." The Gospel of Matthew's Dependence on the Didache. New York: T&T Clark International, 2004, 129-141.
Though the Didache refers in four places to "the gospel," Garrow takes the references to be from a later contribution (Garrow 2004, 129). When comparing 8.2b, 11.3b, and 15.3-4, Garrow, with Niederwimmer, considers the texts to speak of the same thing and to appear to be from the same author (Garrow 2004, 130). Those who reject the idea, according to Garrow, use a "method of redactional analysis [which] is not sensitive to the possibility that one interpolator could make several small insertions across the spread of a text" (Garrow 2004, 130).
There is a valid concern to define "gospel" accurately, as used in the Didache. Garrow notes that an assumption that the gospel is necessarily a written work can cause significant challenges in interpretation (Garrow 2004, 131). Likewise, an assumption of a necessarily oral work can create difficulties. Garrow considers it likely the author of the four references had only one thing in mind, and that it is likely to refer to a relatively stable written text. Only in 8.2b with its comment about the Lord commanding do we have a suggestion that an oral source might be more appropriate (Garrow 2004, 132).
To continue with the concept of a written gospel, Garrow concludes that it must be either Matthew's gospel or some lost text. It is only in Matthew that we find "instructions for every dimension of behavior mentioned by the Didache in connection with 'the gospel'; namely, prayer (Mt. 6.9-13; Did. 8.2b), prophets and visitors (Mt. 10.10, 40-42; Did. 11.3b), church discipline (Mt. 18.15-17; Did. 15.3), and almsgiving (e.g. Mt. 5.42; 6.1-4, Did. 15.4)" (Garrow 2004, 133). Did. 15.3 further calls Matthew 18:15-17 to mind. The concepts are hard to come by elsewhere.
The version of the Lord's Prayer found in Did. 8.1-3 does not precisely match that given in Matthew's Gospel (Garrow 2004, 134). Garrow suggests that the prayer as found in Did. 8.1-3 is of an earlier date than that found in Matthew 6, and that the text about it being the Lord's command in the gospel was a later insertion (Garrow 2004, 136). Given all the data, Garrow concludes tht "the gospel" is a later insertion which refers to Matthew's gospel.
Garrow next asks if 8.2b, 11.3b, and 15.3-4 could have been written by the Modifying Teacher he proposed in an earlier chapter (Garrow 2004, 137). He considers that 8.2b and 15.3-4 would not make sense without the prior existence of material from the modifying teacher layer. If 15.3-4 was written by the modifying teacher, Garrow is left with 14.1-15.4 written by one person. However, he considers the order of introduction of topics to be odd for anything designed by one author (Garrow 2004, 137). It would also be surprising that an author who appeals to the gospel as an authority would then use Malachi 1.11 as an argument for the need for reconciliation, rather than Matthew 5.23-24 (Garrow 2004, 138). Garrow concludes that Did. 14.1-3 did not know of Matthew as an authority, but the author of 15.3-4 did.
Garrow further suggests a multiplicity of authors in Did. 8.1-3 because there is an appeal to the authoritative gospel, but a prayer which is not a quotation from that gospel (Garrow 2004, 139).
Because Garrow sees Didache 11.1-2 as being composed by the modifying teacher, 11.3b, which depends on 11.1-2 for context, would have to be a later insertion (Garrow 2004, 140). The context speaks highly of the ancient practice of the prophets rather than appealing to the authority of the Gospel.
In the end, Garrow concludes the four references to the gospel were later additions which refer to Matthew's Gospel. He also concludes that the bulk of the Didache was written prior to Matthew (Garrow 2004, 141).