On the Kurdish front lines this winter, Peshmerga leaders are repeating a now-familiar story: "We need more weapons," said Captain Ramadan Salah Mohamed from his post on the Khazar frontline, halfway between the Kurdish capital of Erbil and Iraq's second largest city, Mosul, currently under Islamic State (Isis) control.

As his men huddle around a stove burning rubbish, Mohamed describes how a lack of heavy weapons in particular mean coalition air strikes are often the only thing that can stop IS (Daesh) suicide truck bombers from breaking through their lines. They are grateful for the air support and the weapons they have received – particularly the German Milan rockets and G-3 and G-36 assault rifles – but say they need more before advancing further towards Mosul.

"We're in a defensive position here," said Mohamed. "Whether we move forward depends on a political decision."

One political decision made in Washington in late 2014 seemed to promise significant US support to the peshmerga. The 2014 Iraq Train and Equip Fund (ITEF), outlined a plan to train and equip three peshmerga brigades. The first of that weaponry is now tentatively scheduled to start arriving this March but the quantities have been drastically scaled back.

While its provision is seen as a way of securing Kurdish participation in an eventual operation to retake Mosul – a Sunni Arab rather than Kurdish area – critics argue it is insufficient assistance to what they see as the coalition's most reliable force fighting IS.

Ultimately we are waiting for the Kurdistan Regional Government to form these brigades so we can start issuing equipment - Major Thomas Campbell, US Army

In late 2014, after large portions of the Iraqi army dissolved without a fight ahead of IS advances, a hastily assembled international coalition rushed to buttress opposition to the Islamic extremists in Iraq. For many, the Kurds stood out as prime candidates for military support.

The Kurdistan Regional Government (KRG) administered an autonomous region in Iraq's north, which appealed to the west as a relatively stable, democratic and secular ally. Its military – the Peshmerga – proved itself against IS along a 1,050km front line, occupying the oil rich city of Kirkuk when the Iraqi army withdrew and retook the strategic Mosul dam from IS.

By the end of that year, Iran, Germany, the UK, France, Italy and other European nations had gifted the Peshmerga hundreds of rocket launchers and machine guns, thousands of rifles, millions of rounds of ammunition and sundry other equipment.

The US, fully committed to a united Iraq, resolved to channel equipment to the Kurds via Baghdad to avoid angering the federal government or further boost the Kurds' nationalist ambitions. To date, the US has remained guarded about what specific American equipment has gone to the Kurds, preferring to emphasise the role of American air strikes.

When speaking of material support, US officials refer to Washington's contribution to Iraq as a whole, or to the international coalition's support to the Kurds as a whole.

An exception to this is the ITEF, a budget amendment passed by Congress in December 2014, which allocated $1.6bn (£1.1bn) to help train and equip Iraqi forces to fight IS, with $353m of that earmarked for the Peshmerga. The document detailed weapons and equipment for about 15,000 Peshmerga fighters, or three full sized brigades.

The Iraqi army began receiving equipment from the fund in May 2015 but support to the Peshmerga lagged behind. One analyst believes this is because the Iraqi armed forces showed more readiness to retake territory from IS. "The federal government brigades were ready," says Michael Knights, an analyst at the Washington Institute. "They were the ones liberating areas like Tikrit, Baji and Ramadi, whereas the Peshmerga had mostly reached their hold line."

By the time US Secretary of Defence Ash Carter visited Erbil in December 2015, planned support to the Peshmerga had been revised downward to two smaller brigades of 4,000 fighters. The rifles, body armour, machine guns, mortars, vehicles and other equipment would be worth more than $150m, with a further $22m allocated for maintenance and spare parts.

Training would be provided by European coalition partners. Carter also made clear his expectation that the equipment would go to Peshmerga who would participate in an operation to retake Mosul. "That's an important objective," he told reporters, "and precisely our equipment is going to make that possible."

US officials say the original figures in the ITEF were only estimates and that the KRG's Ministry of Peshmerga is responsible for any delays in delivery. "Ultimately we are waiting for the Kurdistan Regional Government to form these brigades so we can start issuing equipment," said Major Thomas Campbell, a US army public affairs officer with the international coalition in Erbil.

Ministry of Peshmerga spokesman Helgurd Hekmat declined to comment on whether the Kurds were responsible for delays in delivery. But analyst Knights says the Kurds were not ready to receive the weapons according to the US stipulations.

"The Kurds could have made this happen faster but they didn't have 4,000 extra peshmerga," he said; new recruits available to be trained and equipped. The Ministry of Peshmerga instead wanted to give the equipment to fighters already on the frontline, which Knights said "is not how the US does it".

Eventually, the US abandoned the idea of creating and training new brigades, with the equipment now destined for the Ministry of Peshmerga's existing number one and two brigades, according to Hekmat.

Knights does not believe the discrepancy between the initial figures proposed in the ITEF indicates a reluctance to support the Peshmerga. "It's pretty notional what they put in it," he said. "It's really just there to show illustratively to congressmen and others what we intend to give; there's a lot of flexibility in there to change the equipment."

Others are not so sure. "I don't understand how scaling this back to two brigades makes sense," said Ernie Audino, a retired US Brigadier General and senior military fellow at the London Center for Policy Research, a New York-based think tank focusing on US security. "The Peshmerga have Mosul isolated on three sides – which is one of the first steps to seizing an objective – they've been patiently occupying that ground for months now waiting for the rest of the Iraq army to get its act together for this fight."

Audino, who spent a year embedded with Peshmerga forces in 2006, says the Kurds need more equipment than what was initially outlined in the ITEF, not less. Referring to the original Kurdish share in the ITEF, he said: "We are allocating 21% to the undeniable main effort in the fight against Isis – that's inconsistent with sound military logic."

In total, the international coalition (which does not include Iran or Russia, which both reportedly support the Kurds) says to date it has provided the Kurds equipment that includes more than 45,000 rifles and machine guns; 56 million rounds of ammunition; 56,000 anti-tank rounds, including 1,000 US AT-4s; 80,000 mortar rounds; and more than 150 vehicles, including 40 American mine resistant ambush protected vehicles. Germany has been the biggest partner, donating equipment including 60 of the much-valued Milan anti-tank rocket launchers and 20,000 assault rifles.

For a military numbering somewhere between 150,000 and 200,000 men, this level of support has been significant. But the Kurds insist they need more and remain ambivalent about advancing further towards Mosul. Hekmat said: "We expect further help for the Peshmerga in the future."