China is a unitary and authoritarian country. Any autonomous region is not autonomous in the sense of a full or partial self-government: all autonomous regions are completely subject to the central government's authority, even on paper.
The autonomous region policy is deliberately adopted instead of self-determination (as opposed to traditional communist ideals) or a federal system (as opposed to the USSR policies, at least on paper), from the argument that, in most places with high concentration of ethnic minorities, the ethnic composition was already diverse with Han and other ethnicities and that a full federal system would harm ethnic relations (I am not going to debate whether this argument is true/good/honest in the context of the question here). On the question of political control, Xinjiang or other autonomous regions cannot be qualified as autonomous, except that some people of ethnic minorities will always be involved in the governance (at least on paper; and like everyone in China they are subject to various pressures).
The autonomy on paper and to an extent in practice is the greater powers compared to other provinces, that the autonomous regions can apply or adopt the general rules, but such modifications may require approval from the central government or may be overturned.
As described in Wikipedia:
The constitution also guarantees a range of rights including: independence of finance, independence of economic planning, independence of arts, science and culture, organization of local police, and use of local language.
These are of course as expected not guaranteed or interpreted as strongly as rights in Western systems, and they are not "independence" but some autonomy, but they still exist to some limited extent. It is helpful to think about the "autonomy" as "may be treated differently than other provinces" instead.
For example, autonomous regions have greater freedom on the taxation policies and do not have to remit budget surpluses to the central government (like other provinces have to). They have different budget policies (e.g. being able to keep greater reserves or budget extra expenses) when the central government is reviewing provincial budgets and determining transfer payments.
The autonomous regions have greater freedom to differ from the general planning on economic and education/science/culture policies, depending on their own particularities. But this is not a full freedom to set these policies without interference from the central government. As a practical example, they may decide to have an additional year of primary education to learn standard Chinese for pupils from ethnic minority backgrounds. The hiring of civil servants and state-owned enterprise employees can have exceptions made for ethnic minorities. There is a guarantee and quota in the administration.
The power to organize police forces is obsolete since the public security administration has been mostly unified (for provinces as well). It was more significant when the policing power was more decentralized.
Despite the limitations and regressions on language rights, ethnic languages are still used by the administration and are still taught in schools (even if reduced). There are still broadcast TV and radio programs in Uyghur, Kazakh, Mongolian and Kirgiz in Xinjiang for example. This is considered an autonomy since the general policy is to promote standard Madarin Chinese.
They may also adapt some generally applicable national laws, subject to approval of the central government. For example, the age of marriage is different for ethnic minorities in Xinjiang (18/20 for women/men instead of 20/22 nationally). In Tibet, existing polygamous marriages were grandfathered. The family planning policies (the former one-child policy for example) were among the most adapted with exceptions for ethnic minorities.
The "autonomy" concerns mostly the bureaucratic details of the administration (which are still important) instead of significant general policy differences, which are unlikely to be tolerated.
Xinjiang has also a complication that a parallel political administration (the Corps) exists for later mostly Han settlers from other parts of China in many areas in Xinjiang. Within the population under the Corps' administration, ~90% are Han (but not all Han is under the Corps: it is organized territorially not ethnically). The Corps have the relative autonomy from the Xinjiang Autonomous Region, which reduces the control of the latter over some of its territory.
