k-means with one cluster K-means may give different results, because the initial choice of centroids is random. 

However, if I were to choose k=1, will the algorithm always provide the same answer equal to the "barycentre" of my data?
Yes. The centroid will converge to the center of all your data and this will occur in a single iteration. This is due to all the data points belonging to a single centroid, thus it will be centered according to all these instances immediately. 

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans

# Generate data
class1 = np.random.randn(1000, 2)

class2 = np.random.randn(1000, 2)
class2[:,0] = 4+class2[:,0]
class2[:,1] = 4+class2[:,1]

class3 = np.random.randn(1000, 2)
class3[:,0] = -4+class3[:,0]
class3[:,1] = 4+class3[:,1]

data = np.append( class1, class2, axis= 0)
data = np.append( data, class3, axis= 0)
print(data.shape)

# Plot the data
plt.scatter(data[:,0], data[:,1])
plt.show()

# Cluster
kmeans = KMeans(n_clusters=1, random_state=0, verbose = 1).fit(data)

# Plot clustered results
plt.scatter(data[:,0], data[:,1], c=kmeans.labels_)
plt.scatter(kmeans.cluster_centers_[:,0], kmeans.cluster_centers_[:,1], c = 'r')
plt.show()

# Show the cluster centers
print(kmeans.cluster_centers_)





  Initialization complete Iteration  0, inertia 81470.055 Iteration  1,
  inertia 48841.695 Converged at iteration 1: center shift 0.000000e+00
  within tolerance 8.140283e-04